Recall that a sequence ( ) converges to L if and only if

Prove that if then .

Attempt:

Suppose , then

Then we know

The limit of the two outside terms is L, and hence the limit of is L by the squeezing theorem for sequences.

Conversely we can say: suppose then

etc...

But I doubt this is right or even the right approach. Can anyone show me a better and more rigorous proof?