Sequence Proof (Am I missing something here?)

[Newtime]

Sequence Proof (Am I missing something here??)

The reason I'm posting this is because I just took an exam and this was one of the questions, and it was so easy I feel like I may have been completely overlooking a complicating factor:

-{b_n} is a sequence of positive real numbers. prove that either it contains a convergent subsequence or converges to positive infinity.

proof.

two case: either the sequence is bounded or unbounded. if it is bounded, apply the bolzano-weirstrass theorem to conclude that it contains a convergent subsequence. if unbounded, by definition, the sequence goes to positive infinity.

qed

On the exam I used better notation and wording but that's it essentially. so what's the consensus...is this valid?

 

[l'Hôpital]

lol nvm. Misread.

 

[mathman]

The reason I'm posting this is because I just took an exam and this was one of the questions, and it was so easy I feel like I may have been completely overlooking a complicating factor:

-{b_n} is a sequence of positive real numbers. prove that either it contains a convergent subsequence or converges to positive infinity.

proof.

two case: either the sequence is bounded or unbounded. if it is bounded, apply the bolzano-weirstrass theorem to conclude that it contains a convergent subsequence. if unbounded, by definition, the sequence goes to positive infinity.

qed

On the exam I used better notation and wording but that's it essentially. so what's the consensus...is this valid?

The unbounded case is more complicated. It might have a subsequence which "converges" to infinity, as well as another subsequence which is convergent to a finite number.