# recurrent sequence in a compact space

please how to solve this question

Let $$E$$ a compact metric space and a function $$f:E\to E$$ such that $$\forall x,y \in E, d(f(x),f(y))\geq d(x,y)$$ Let $$a\in E$$ and a sequence $$(f^n(a))$$ , prove that $$a$$ in an adherent value of $$(f^n(a))$$.

As E is compact then $$(f^n(a))$$ has a convergent subsequence, but I stop here

Hint: Prove by contradiction. Suppose $$(f^n(a))$$ stays away from $$a$$, so there is some $$\varepsilon>0$$ such that $$d(f^n(a),a)>\varepsilon$$ for all $$n\geq 1$$. Prove that $$(f^n(a))_{n>m}$$ also stays away from $$f^m(a)$$ for all $$m$$ (with the same $$\varepsilon$$), and show this yields a contradiction with compactness of $$E$$.
• That's why you need the whole sequence avoids being arbitrarily near $a$, so there is no way a subsequence converges to $a$. – user10354138 Jun 12 at 16:01