Introduction to Limits

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This lesson will introduce the notion of a limit.

Definition (sequence)

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A sequence on a topological space   is a function  . Alternatively, it is a list   where   for all  . The sequence is often denoted as  

Definition (limit of a sequence)

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The point   is a limit of the sequence   if for every neighborhood  , there is   such that   for all  . In this case, we write   and say that   converges to  .

Note that a sequence might have multiple limits. For example, in any space with the indiscrete topology, every sequence converges to every point of the space!

Example

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Let   be the set of integers with the topology where   is open if   (called the finite complement topology). Let  . Then we see that   for any  . Indeed, note that given any neighborhood  ,   contains all but finitely many points of  . Let   be the maximum of all of the numbers not contained in  . Then for all  , we see that  , hence  .

Theorem

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If a space   is Hausdorff and the sequence   in   has a limit, then that limit is unique.

The proof of this theorem is left as an exercise to the student. The hint is to assume that there are two distinct limits and show that this leads to a contradiction.

Theorem

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If   is a continuous function and   in   then   in Y.

Proof

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Let   be an open neighborhood of  . Since   is continuous,   is open in   and, by definition, contains  . Therefore, there is   such that   for all  . Therefore,  . Thus,  .

The converse of this theorem is, in general, false. However, it is true for metric spaces. (In fact, it holds for any first-countable space.)