Two proofs regarding limits

I just want to make sure I have done these proofs correctly.

1) Let be a bounded sequence such that there exists such that for all , and let be a sequence such that . Prove .

PROOF:

Assume the given conditions above. This means that there exists such that, if and , then . In particular, there exists such that if and , then . Hence, , given that and . And so, . Q.E.D.

2)Suppose that and are sequences such that for all and . Prove .

PROOF:

Assume the given conditions above. This means there exists such that, if and , then . Hence, , and so there exists such that, if and , then . And so, . Q.E.D.

Any feedback would be appreciated, thanks.