1. ## Question

There is a question that I have been having hard time finding a formal solution. It's an easy question I don't know why I have been having hard time solving it, but here it is.

(an) is a non decreasing sequence.
Prove that if the sequence is not bounded then it converges to infinity.

So first I tried to assume that it doesn't converge to infinity and somehow get a contradiction to the fact that it's a non decreasing sequence or that it is bounded but I failed.

So I said that , if the sequence is not bounded that means for every M in R (Real numbers) there exists an N such that M<aN
It's a non decreasing sequence therefore for every n in Natural numbers that n>N we have M<aN<an So we get that for every M in R there exists an N such that for every n>N we get M<an and therefore it satisfies the definition of the limit (for infinity) Therefore it converges to infinity.

So can we count what I wrote as a proof?

My second question is, if it is correct, what other ways could I have used to prove? I got a little bit confused with the contrapositive here. Could I have assumed that it doesn't converge to infinity and somehow get a contradiction?

Thank you.

2. ## Re: Question

diverges to infinity. sequences don't converge to infinity.

Other than that wording your proof looks correct.

For your second question the answer is yes. If you assumed that $a_n$ converged to some value you could show that contradicts $a_n$ being unbounded but there is really no need for that here. Your proof is essentially 2 lines.