Three questions, hope that's not too many!

1. I need to prove that the sequence

What I have is for any there exists such that

I take that as

and get

Here is my first question: when I put that into

I need a fixed vaule for . What am I missing?

2.

Prove that every unbounded sequence contains a monotone subsequence that diverges to infinity.

On this one, I know unbounded can mean divergent. What do I do next to prove this? I am confused on this one!

3.

Prove that if a sequence converges to 0, and a sequence is bounded, then the sequence converges to 0.

I had gotten help on this before, but hadn't gotten very far. I'm starting with

, but how would I put this into a proof for this? again, confused!

Thanks in advance, and I might have some more later!