Formal definition for limit of a sequence
what i want to do in this video is to provide ourselves with a rigorous definition of what it means to take the limit of a sequence as n approaches infinity and what we'll see is actually very similar to the definition of any function as a limit approaches infinity and this is because the sequences really can be just viewed as a function of their indices, so let's say let me draw an arbitrary sequence right over here so actually let me draw like this just to make it clear but the limit is approaching so let me draw a sequence let me draw a sequence that is jumping around little bit, so lets say when n=1, a(1) is there, when n = 2, a(2) is there, when n = 3, a(3) is over there when n=4, a(4) is over here, when n=5 a(5) is over here and it looks like is n is so this is 1 2 3 4 5 so it looks like that as n gets bigger and bigger and bigger a(n) seems to be approaching, seems to be approaching some value it seems to be getting closer and closer, seems to be converging to some value L right over here. What we need to do is come up with a definition of what is it really mean to converge to L. So let's say for any, so we're gonna say that you converge to L for any, for any ε > 0, for any positive epsilon, you can, you can come up, you can get or you can, there is let me rewrite it this way, for any positive epsilon there is a positive, positive M, capital M, such that, such that if, if, lower case n is greater than capital M, then the distance between a(n) and our limit, this L right over here the distance between those two points is less than epsilon. If you can do this for any epsilon, for any epsilon, greater than 0, there is a positive M, such that if n is greater than M, the distance between a(n) and our limit is less than epsilon then we can say, then we can say that the limit of a(n) as n approaches infinity is equal to L and we can say that a(n) converges, converges, converges to L. So let's, let's, let's parse this, so here I was making the claim that a(n) is approaching this L right over here, I tried to draw it as a horizontal line. This definition of the of what it means to converge for sequence to converge says look for any epsilon greater than zero. So let me pick an epsilon greater than zero, so I am gonna go to L plus epsilon, actually let me do it right over here, L, so see this is L plus epsilon and let's say this is right here this is L minus epsilon. So let me draw those two bounds, right over here. And so I picked an epsilon here so for any, for any arbitrary epsilon, any arbitrary positive epsilon I pick, we can find a positive M, we can find a positive M, so that as long as, so let's say that is our M right over there. So that as long as our n is greater than our M, then our a(n) our a(n) is within epsilon of L, so being within the epsilon, being within epsilon of L is essentially being in this range. This right over here is just saying, look that the distance between a(n) and L is less than epsilon, so that would be any of these, anything that is in this, between L minus epsilon and L plus epsilon. The distance between that and our limit is going to be less than epsilon. And we see right over here, at least visually, if we pick M there and if you can take an n that's larger than that M, if you pick an N that's larger than M, if M is equal to 3, a(n) seems to be close enough. If M is 4, a(n) is even getting closer. It's within our epsilon. So if we can say, if we can say that it is true, for any epsilon that we pick, then we can say, we can say that the limit exists, that a(n) converges to L. In the next video we will use this definition to actually prove that a sequence converges.