Proving bounded monotonic sequences must converge

Homework Statement

I'm approaching this problem from a different method than conventially shown.

Homework Equations

if lim=infinity for all M>0, there exists a N such that n>N => {s(n)}>=M

The Attempt at a Solution

this can be rewritten as:

{s(n)} is a sequence. If {s(n)} is bounded and monotonic , then {s(n)} converges.

the contrapositive is,

{s(n)} is a sequence. If {s(n)} doesn't converge then it is either not monotonic or not bounded.

Hence, if I just show that it is not monotonic, then the proof works.

PROOF:

{s(n)} doesn't converge. So if {s(n)}=+infinity, then that means that for all M>0, there exists an N such for all n>N, M>={s(n)}.

Then {s(n)} is not bounded.

Thus all monotonic sequences must converge. Q.E.D

Does this proof work?

1)I'm a bit worried because I don't even know if I can apply contrapostives this way.

2)I don't know if I have to show {s(n)} is not bounded for all cases of {s(n)} diverging(i.e +infinity, -infinity, DNE). It seems plausible that if I can show that {s(n)} is not bounded when {s(n)} diverges for even just one of the cases, then the theorem works.
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