IMC 1996 · Day 2 · P7

The “only if” part is obvious. Now suppose that $\lim\limits_{n \to \infty} (x_{n+1} - x_n) = 0$ and the sequence $\{x_n\}$ does not converge. Then there are two cluster points $K < L$. There must be points from the interval $(K, L)$ in the sequence. There is an $x \in (K, L)$ such that $f(x) \ne x$. Put $\varepsilon = \dfrac{|f(x) - x|}{2} > 0$. Then from the continuity of the function $f$ we get that for some $\delta > 0$ for all $y \in (x - \delta, x + \delta)$ it is $|f(y) - y| > \varepsilon$. On the other hand for $n$ large enough it is $|x_{n+1} - x_n| < 2\delta$ and $|f(x_n) - x_n| = |x_{n+1} - x_n| < \varepsilon$. So the sequence cannot come into the interval $(x - \delta, x + \delta)$, but also cannot jump over this interval. Then all cluster points have to be at most $x - \delta$ (a contradiction with $L$ being a cluster point), or at least $x + \delta$ (a contradiction with $K$ being a cluster point).