I was going over my notes and came across the following theorem:
Suppose $(a_{n})$, $(b_{n})$ are real sequences of positive real numbers. If the following hold:
(i) $(b_{n})$ is convergent;
(ii) $(a_{n})$ is increasing; and
(iii) $a_{n} \leq b_{n}$ for all $n\in\mathbb{N}$,
then $(a_{n})$ is also convergent.
I have followed the proof, which seems fairly straightforward: $(b_{n})$ is convergent so bounded above. From (iii) we can then deduce that $(a_{n})$ is also bounded above. Since $(a_{n})$ is bounded above and increasing it converges to its supremum and the result follows.
But it doesn't seem to use the positivity requirement of the sequences. Why do $a_{n},b_{n}>0$ for each $n\in\mathbb{N}$? Or am I missing something?
Thanks.