This is a question that seems obvious except my method doesn't look quite right.

Here's what i've done so far:

N.T.S (need to show)

converges.

:

Adding these two expressions gives:

From here i'm not sure where to proceed. I'm not even sure if this is the right way of doing it!

I have noticed that I can get all the terms except for

. However, I can't see a way of connecting the two subsequences apart from adding them together.

I was thinking of making

and then I would have a nice increasing sequence that has a limit a. Otherwise i'm having some difficulty in seeing what to do next.

Could someone please give me a nudge in the right direction? I would be grateful if they could!

EDIT: I've just noticed that you can get a_1 if youmake n=0 for the second subsequence. I can't actually see how this helps unless I can somehow see the property that a_{2n+1} is less than a_n. Even then I would have trouble justifying the order a_1<a_2_a_3......etc