 So, what we will do is see as we go into the subject we will need a little we will need keep needing concepts from some of the other branches of mathematics. So, I will introduce some of these concepts in the beginning. We will also tell you what it means for a solution to exist and when is there is a solution and so on. So, you have to thankfully there are some blanket theorems that that we can apply which guarantee the existence of a solution. So, you have to make sure that your problem the problem you have at hand satisfies the assumptions of those theorems and then you have existence for free no problem. You do not have to so this the existence question can become a chicken and egg problem because you are the existence should not require you to claiming the existence should not require you to actually compute what you are for opening exists. So, you should have a way of claiming existence that does not require finding the solution in the first place. So, thankfully there are such results so we will go through those. So, we will start with this. So, first simple idea we need is that of a so I will start with just a review of real analysis and later also some convexity. So, the first concept we need is that we need is that of a sequence. In this in this course we will stick to problems that are in Euclidean spaces so in Rn a lot of what I will teach you can also be applied in larger spaces but we will mostly stick to Rn. So, I will talk of a sequence here which is in Rn. So, the way I will write this is so you have a point X that lies in Rn. Now, you have and a sequence is going to is just a series of such points. So, this is so X. So, here is a point X 1 has another point X 2 etc. X 3 etc. So, a sequence is denoted like this X sub K with and on outside the bracket you write K in N. So, you have so a sequence is an infinite series of points X 1, X 2, X 3 etc etc all lying in a common space. So, all of them say suppose belong to Rn. Now, since you are in Rn we can talk of how far is this sequence there is a certain sequences have a certain direction in which they go as K increases you can sort of think of them as a moving particle or whatever or a fly that is flying around in this room. So, at time 1, at time 2, at time 3 etc where is this particle currently we can this is how you can track a sequence and so your since we are in Rn we can so here is your the origin of your space and we can talk of how far is where is this sequence added. So, as first we say that the sequence is bounded you say it is bounded if you have the following. So, if you look at this particular figure this sequence is traveling all over the place the indices keep going on and on till infinity the sequence keeps traveling. We say it is bounded if there is some big ball centered at the origin if there is a ball centered at the origin in which it in which the entire sequence resides. We say Xn is bounded if there exists an m let us say greater than 0 such that if I take them that is norm of Xk norm of Xk is nothing but the distance of Xk from the origin norm of Xk is less than equal to m for all k in natural number. So, we say it is bounded if it is if the sequence if the norm of X of every point in the sequence is no more than m there is a so important notice the sequence the order in which this is being stated. We want a common m that will work for all k we want there exists an m such that for all k norm of Xk is less than equal to m. You can of course find an m that is tailored to a particular to each point. So, then that does not make the sequence itself bounded that just means that every point has a finite would have a finite norm. So, here we are asking for a common m such that the norm of Xk is less than equal to m for all k. So, the picture you should have in mind is that there is this ball of radius m here in which this entire sequence is lying of course the ball is not necessarily minimal or in any sense it is just some ball of finite radius. It is not minimal in any sense I mean that it does not mean that you cannot find a smaller ball or anything like that but all that matters is that the entire sequence is within the ball in that we say it is bounded. Now I said that we can follow the sequence along its progression or as k increases and you can ask okay where is this sequence headed. So, simple answer is firstly is it at least not running away eventually if is it not running away for example if you have a sequence Xk say for example in R let us take a sequence in R say Xk equal to k square such a sequence what would happen to such a sequence this is a sequence that keeps increasing as k increases eventually if you look at it along the real line here is your 0 of the real line at k equal to 1 it is at 1 at k equal to 2 it is gone to 4 k equal to 3 it is gone to 9 and so on so eventually it will just it will it keep blowing up increases and increases. There is no ball m that you can specify such that all the elements of the sequence are within m right within absolute value less than m okay. So, this is this sort of sequence is not bounded let us take another example consider the sequence Xk equal to minus 1 to the k minus 1 raise to k how does this sequence behave it oscillates between plus 1 and minus 1 right. So, at k equal to 1 it is at k equal to 1 it is at minus 1 at k equal to 2 it is at 1 k equal to 3 it goes back to minus 1 4 at 1 etc etc is this sequence bounded right it is bounded it is bounded because you can I can put all these all the elements of that sequence or can all be enclosed within this plus minus 2 region right. So, I can take say for example m equal to 2 then and then the absolute value of Xk or the norm which is the absolute value of Xk would be less than equal to it. So, this sequence is this sequence is bounded now just because it is bounded does not mean the sequence is headed somewhere sequence could still dance around all the time without without having any particular particular direction but interesting sequences are those with are those and which we are always interested in are these sequences that that may initially dance around for a long time but eventually head somewhere right what this is saying is so we say that Xk converges to X star if the following is true you tell me you specify a distance you specify how what it means to be how what it means to be close to X star and well you specify it by saying okay well I want I wanted to be within the distance between X and X star to be no more than epsilon use this you specify the epsilon. So, for every epsilon that you you specify now it is for me a challenge I have to now come tell you does the sequence is the sequence going to be in within within epsilon of X star unfortunately usually it is not the case that the entire sequence is going to be within epsilon of X star but then maybe not the full sequence but does the sequence eventually end up within within epsilon of X star within epsilon distance of X star what does it mean for it eventually to end up so you for every epsilon that you throw as a challenge to me I should be able to produce an N naught I should be able to produce an N naught such that from the point N naught onwards the sequence completely lies within a within an epsilon distance of X star so from epsilon or from N naught onwards I am within this so the picture you should have in mind is so again let us draw pair of axes here is your origin here is the sequence initially it can do whatever it wants maybe and eventually it has this behavior where it seems to be tending towards some point and that is it I cannot even draw this more finely here somewhere here is the X star it is headed here and then eventually after a point all the all the elements of the sequence are accumulating around X star and eventually since for every epsilon I should be able to find this find an N naught like this it means that however small you make the epsilon infinitely many of points of the sequence are actually within epsilon of X star within epsilon distance of X star only the first N naught are going to be outside right and however small I make the epsilon means I can make the epsilon 10 raise to minus 10 and still within that are going to be infinitely many points and outside only finitely I can make it 10 raise to minus 20 still within that are going to be infinitely many and outside are going to be only finitely okay so form was there will be a point onwards from which it is going to be inside right so let us take another let us take an example let us take X k equal to 1 by k okay again a sequence of real numbers X k equal to 1 by k what does this does this converge and it does if it converges converges to what converges to 0 why does it converge to 0 as see say as k increases it becomes smaller and smaller that is while that is true we want to say we need to say it based on this definition based on the definition that I have stated so for every epsilon greater than 0 can you find for me an N naught can you find for me an N naught such that for beyond that beyond N naught all all the X k's for k greater than N naught have absolute value less than epsilon yeah precisely so far so what I can do is so what I need is that this I need to find an N naught so you are giving me an epsilon giving me an epsilon which is positive and I need to find an N naught what will I take my N naught as what I can do is I can take N naught as as I can I can do this right so I can I can take N naught as 1 by epsilon and and just seal that to the to the greatest integer sorry the smallest integer greater than 1 by epsilon then what would happen from so X N naught itself is going to be definitely less than X N naught is going to be definitely less than less than equal to 1 by epsilon or I can make this plus 1 so this is definitely less than 1 by epsilon X N naught is less than 1 by epsilon and not only X N naught X N naught plus 1 X N naught plus 2 etc all of them are so not 1 by epsilon sorry 1 epsilon sorry final state X N naught is going to be less than epsilon and not only that X N naught plus 1 X N naught plus 2 X N naught plus 3 all of them are going to be less than 1 by epsilon right so in short the sequence as you can see is going to it may it may have a large number of values that are away from 0 but infinitely many form a certain point onwards all of them are going to be as close to 0 as you like right so X star in this case is called the limit of the sequence it is called the limit of this sequence X k and it is very easy to show that limit of a sequence if it exists you cannot have a sequence that converges to two different points I did not specify this generally it is not needed because we are talking of distances between points between X k and X star no no no but then they have to if you have a point in a different space then no no you need to you need to embed this all of them in the same space so a distance like this is specific is meaningful provided both distance between two points is meaningful provided they both have the same number of coordinates at least so both must have n coordinates in this case okay so we talked of so what it means to converge so limit of a sequence if it exists is unique okay can you tell me what it means for a limit to not exist when does a when does a sequence not have a limit that does not say that the limit does not exist that just says that the limit is not X star so let us first do the simple thing what does it mean for a sequence to not converge to a given point X star it means that there exists so so let me write the simpler version so the answer to this is there exists greater than 0 such that for all n not in natural numbers X k minus X star is greater than equal to epsilon for some k greater than what does this mean so when a sequence is not converging to an it may converge but it is certainly not converging to this particular X star that I am considering okay when a sequence is not converging to this particular X star it means that I can find an epsilon ball around that X star and epsilon radius region around that X star such that no matter where you start no matter where which n not you start from there is always going to be at least one k after that which escapes from that from that region there is going to be one k after that n not such that X k escapes from that is this clear so what it what this means is so the way this looks is so here is suppose my X star I can I should be able to find some epsilon region around it okay so here is this epsilon radius so and the way the sequence will behave is that sequence may come inside goes outside goes outside once may be come inside again goes outside again come inside again but goes outside again etc etc point is it never eventually lies inside so it keeps going out infinitely infinitely many times so you try you look at the sequence from any n not onwards you will always find one point from there onwards which is escape from any n not onwards no matter how large your n not so you can never find an n not such that the sequence from here onwards is forever inside okay so there is such a sized ball okay for which this for okay and usually this such a so to make this true the ball has to become would have to become very very small so this will become easier and easier if the ball is smaller because it will becomes easier for the sequences points to escape from it right if you throw your net very wide obviously many many points will lie inside but you there exists such a such an epsilon that is the that is very that becomes easier to show once your epsilon becomes smaller okay so the sequence keep escapes escaping like this okay now if you look at this this these points that have escaped okay so for every n not I can find one x k with k greater than n not that has escaped right so let me just call that something so let me call that let me call that this particular k this k let us just index it by let us call it k n not k sub n not okay so what it means is this there is always an x k n not which is gone out right and not only that there is such a x k n not for every n not right for every n not so this itself these points this this set of points itself can be thought of as a sequence so they are they are traveling this is a so this here is a an example of a subsequence of x k of the original sequence so a subsequence a subsequence is formed like this you what you you have your you have your sequence x 1 x 2 x 3 etc the way you construct a subsequence is by construct you take a subset of the indices so you take k n you consider a sequence of natural numbers like this you consider these k 1 k 2 dot these are natural numbers such that this k i plus 1 is strictly greater than k i right so when you index the index the subscript here you the natural number jumps by a positive amount okay k i plus 1 is strictly greater than then you can with this once I have this so this is now a strictly increasing sequence of natural number I can use these as indices on on my original sequence and that gives me a subsequence see if I so this is just a subtle technical point if I did not have this this this point here where I am asking for strictly greater right the sequence could go backwards also so that changes the direction of moment motion of the sequence so we do not allow that in in the definition of the subsequence if you have equality the sequence stalls although i is getting index k i remains the same so essentially you are taking repeating the same point from the original sequence and we do not want to allow that we want the subsequence to move alongside the sequence okay and in the same direction so k 1 k 2 k 3 are just indices these are natural numbers these are just some natural numbers so I am going to use these to index k so to index my original sequence I am going to look at the sequence at k at at k 1 k 2 k 3 etc for example k k 1 could be 2 k 3 k 2 could be could be 7 suppose etc so as I pull this as I put look at the sequence with these indices only what I get is a subsequence of the entire sequence so we will stop here now we will continue this in the next class