TITLE: Does an increasing sequence of reals converge if the difference of consecutive terms approaches zero?
QUESTION [35 upvotes]: 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?

REPLY [4 votes]: Note that if we define $b_n=a_{n+1}-a_n$, then $a_n=a_0+\sum_{n=0}^{\infty}b_n$. So this question is equivalent to asking whether the terms of an infinite series going to zero is sufficient for the series to converge. There are a variety of examples of series with terms that go to zero, yet do not converge, with the harmonic series ($\sum \frac 1 n$) being one of the most famous. 
And in fact we can construct a counterexample from any sequence by defining a sequence $c_n$ by simply re-indexing the terms. We set $c_0$ equal to $a_0$. Then set $c_{k1}$ equal to $a_1$, where $k_1>a_1-a_0$, and fill in the terms $c_1$ to $c_{k-1}$ with equally spaced terms; this will result in all of the consecutive differences from $c_0$ to $c_{k1}$ being less than $1$. Then set $c_k2$ equal to $a_2$, where $k_2+k_1>2(a_2-a_1)$, which results in consecutive differences between $c_{k1}$ to $c_{k2}$ being less than $\frac 1 2$. Just keep re-indexing each term and filling in more and more new terms, and you can drive the consecutive differences arbitrarily low.
Another approach is to look at a sequence as an approximation of a continuous function, and the difference between successive terms as an approximation of the derivative. Then we just need a function such that $f'(x)$ converges to zero, but $f$ diverges. Two examples of such are the log function, (which gives a sequence very similar to the harmonic sequence) and square root. Note that both of these examples can be obtained by taking the inverse of a function whose derivative is constantly increasing. If $g'$ goes to infinity, then $(g^{-1})'$ goes to zero. But if the domain of $g$ is the whole real line, then the range of $g^{-1}$ is the whole real line, i.e. $g^{-1}$ goes to infinity.