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.