# how to show that f has only one fixed points?

Let (X, d) be a complete metric space. Let$f : X → X$ be a function such that for all distinct$x, y ∈ X$ ,

$d(f^ k (x), f^ k (y)) < c · d(x, y)$, for some real number $c < 1$ and an integer $k > 1$. Show that f has a unique fixed point.

my attempt : i take $f(x) = x$and $f(y) = y$ .now $f^k = f(f(...(x),,))))= x$ similarly $f^k =f(f(,,,,,(y))..) = y$