I'm not sure how much it would have helped here (because I think some other things went wrong at the same time!), but in general the kind of explanation you've just given makes things *so* much easier for your helpers to follow...
This isn't sufficient. In particular your treatment of "the former part" is insufficient.
The question explicitly tells you that "you will need to say something about the convergence of
2NN as
N→∞". Moreover, it gives you a
hint (and so you can't simply say "I considered it and it tends to 0"
.
[Also, I'll note that 1+(1/x) "gets closer and closer to 0" as x goes to infinity, but it sure as heck doesn't tend to 0...]
On the other hand, I can't say I find their hint terribly useful.
If I wanted to prove
N/2N→0 I would go (there are details in the following you'll need to fill and, as well as all the bits where I've just written "..." rather than give the game away):
Let
an=n/2n. Then
anan+1=... So if n > ... then
anan+1<43. Then by comparison with the geometric series AR^n (with A = ..., R = ...), a_n tends to 0.
As I say, I really don't get the hint. (I can think of ways to use it, but they feel inappropriate given the rest of the Q).