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Current time:0:00Total duration:4:49

Formal definition for limit of a sequence

Video transcript

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 as n approaches infinity and what we'll see is it's actually very similar to the definition of any function of taking any function as a limit approaches infinity and this is because these sequences really can be disputed as a function of their indices so let's say let me draw let me draw an arbitrary sequence right over here so actually we draw it like this just to make it clear what the limit is approaching so let me draw a sequence let me draw a sequence that is jumping around a little bit so let's say when n is equal to 1 a sub 1 is there when n is equal to 2 a sub 2 is there when n is equal to 3 a sub 3 is over there when n is equal to 4 a sub 4 is over here when n is equal to 5 a sub 5 is over here and it looks like as n is so this is 1 2 3 4 5 and so it looks like as n it gets bigger and bigger and bigger a sub n seems to be approaching seems to be approaching some value it seems to be getting closer and closer it seems to be converging to some value L right over here but what we need to do is come up with a definition of what does it really mean to converge to L so let's say for any so we're going to say that you converge to L if for any for any epsilon greater than 0 for any positive epsilon you can you can come up you can get or you can there is when we're going to write this way for any positive epsilon there is a positive positive m capital M such that such that if if lowercase n is greater than capital m then the distance between a sub n a sub n and our limit 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 zero there is a positive M such that if n is greater than M then the distance treat a sub N and our limit is less than Epsilon then we can say then we can say that the limit of a sub n as n approaches infinity is equal to L and we can say that a sub n converges converges converges to L so let's let's let's parse this so here I was making the claim that this that a sub n is approaching this L right over here I try to drive draw it as a horizontal line this definition of 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'm going to go to L plus Epsilon actually let me do it right over here L let's say this is L plus Epsilon and let's say this is right here this is L minus epsilon L minus Epsilon so let me draw those two bounds right over here and so I picked an epsilon here so for any if for any arbitrary absolute 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 that is our M right over there so that as long as our n is greater than our M then our a sub n our a sub n is within epsilon of L so being within epsilon being within epsilon of L is essentially being in this range this right over here is just saying look the distance between a sub N and L is less than Epsilon so that would be any of these anything that's 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 if you take an N that's larger than that M if you pick an N that's larger than M if n is equal to 3 a sub n seems to be close enough if n is for a sub n is even getting closer it's within our epsilon so if we can if we can say this is true for any epsilon that we pick then we can say we can say that the limit exists that a sub n converges to L in the next video we'll use this definition to actually prove that a sequence converges