Let $(M,d)$ be a compact metric space, and $f:M\to M$ be bijective and satisfy $d(f(x),f(y))\le d(x,y)\ \ \forall x,y\in M$.
Now, if $x_0 \operatorname{:= }x, x_n \operatorname{:= } f(x_{n-1})$, then show that there exists a convergent subsequence of $(x_n)$ converging to $x$.
I was able to get a convergent subsequence above, but I wasn't able to show that it converged to $x$.