$f \in \mathcal{C}(X, X)$ on metric space with $\sum_{n=1}^\infty d(f^n(x), f^n(y)) < \infty$ has a fixpoint

Question

Let $X$ be a complete metric space and $f : X \to X$ continuous such that $$\sum_{n=1}^\infty d(f^n(x), f^n(y)) < \infty $$ for all $x, y \in X$, where $f^n$ means $f \circ \ldots \circ f$ $n$-times. Then $f$ has a fixpoint.

Since the sum convergence, for all $x, y \in X$ it's true that $d(f^n(x), f^n(y)) \longrightarrow 0$. How can I go from here? The fact that $f$ is continuous is essential, isn't it?

Answer 1

Hints:

1. What does it mean for $a$ to be a fixed-point of the mapping $f$?
2. To prove two things are equal, it's enough to prove their distance is zero.
3. Yes, continuity is essential.
4. There's another hypothesis that you have not used yet.

Answer 2

Hint: show for any $x$, $f^n(x)$ is a Cauchy sequence.