If $a_n$ is a sequence such that
$$a_1 \leq a_2 \leq a_3 \leq \dotsb$$
and has the property that $a_{n+1}-a_n \to 0$, then can we conclude that $a_n$ is convergent?
I know that without the condition that the sequence is increasing, this is not true, as we could consider the sequence given in this answer to a similar question that does not require the sequence to be increasing.
$$0, 1, \frac12, 0, \frac13, \frac23, 1, \frac34, \frac12, \frac14, 0, \frac15, \frac25, \frac35, \frac45, 1, \dotsc$$
This oscillates between $0$ and $1$, while the difference of consecutive terms approaches $0$ since the difference is always of the form $\pm\frac1m$ and $m$ increases the further we go in this sequence.
So how can we use the condition that $a_n$ is increasing to show that $a_n$ must converge? Or is this still not sufficient?
Best Answer
An easy way to visualize why this can't be true is to try putting some points on a number line.
Start with 1 point in [0, 1):
2 points in [1, 2):
And so on:
Now you have a sequence that grows to infinity but keeps getting closer together.