# Shrinking Map and Fixed Point via Iteration Method

Let $T$ be a map from a compact metric space $X$ into itself satisfying $d(Tx,Ty) < d(x,y)$ for all distinct $x,y$ in $X$. It is true that $T$ has a unique fixed point. Fix $x_0 \in X$ any point, and define the recursive sequence $\{ x_n \}$ by $Tx_0 = x_1$, and $T^{n} x_0 = x_n$. If this sequence converges, it is true that it converges to the unique fixed point in $X$. However, I am having trouble showing that the sequence does converge, i.e., is Cauchy. I can show the sequence is Cauchy if the mapping had been a contraction, but the technique there doesn't generalize well.

Edit: I came up with a simpler proof than from anything else I could find online, but it's not quite complete. Suppose for the sake of contradiction that $\{T^nx_0\}$ did not converge to the fixed point $\alpha$. Then $\exists \epsilon$ and a subseq $\{ T^{n(k)} x_0 \}$ s.t. for each $n(k)$, $d(T^{n(k)} x_0 , \alpha ) > \epsilon$. By compactness, $\{ T^{n(k)} x_0 \}$ admits a convergent subseq. I want to say this subseq, say $\{ T^{n_k(m)} x_0\}_{m \in \mathbb{N}}$ and converging to $\beta$, must actually converge to $\alpha$ by continuity of $T$, but can can this be done? It's true that $T(T^{n_k(m))} x_0) \rightarrow T(\beta)$, but can I say $T^{n_k(m) + 1} x_0 \rightarrow \beta$, so that $\beta$ is actually the fixed point $\alpha$?

• I think this will help jstor.org/pss/2044417 – Norbert Feb 13 '12 at 20:40
• What's $T$? Just some random continuous map from $X$ to itself? Or is related to $f$ somehow? – Jason DeVito Feb 13 '12 at 21:12
• @JasonDeVito Sorry, $T$ is $f$! I edited the post. – Peter Feb 13 '12 at 21:28