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## Homework Statement

Given that limit of [itex]s_{2n}[/itex] is L and limit of [itex]s_{2n+1}[/itex] is L, prove that lim [itex]s_{n}[/itex] is also L.

## Homework Equations

## The Attempt at a Solution

This seems very obvious: If the even terms of a sequence approach a number and the odd terms of that sequence approach the same number, then the sequence itself approaches that number.

But I'm not sure how to go about translating this into mathematics. I know from the definition of a limit that I can make the odd and even terms of [itex]s_{n}[/itex] as close to L as I want given a large enough n, but what I really need is to go from that to

given e>0 there exists natural number N so that n > N implies |[itex]s_{n}[/itex]-L|<e