Assume we have a continuous map $f:\mathbb{R}^n\to\mathbb{R}^n$, and there exists at least one point $a\in\mathbb{R}^n$ satisfies the sequence $\{a,f(a),f\circ f(a),\cdots\}$ is bounded in $\mathbb{R}^n$. Can we prove that map $f$ has at least one fixed point?
I think we should use Brouwer fixed-point theorem, since the sequence is bounded we can find a closed ball cover the sequence. But how can i find such a closed ball $B$ satisfies $f(B)\subset B$? Any help would be appreciated!
Consider $$(x,y,z)\mapsto (y,-x,z+x^2+y^2-1) $$ and $a=(1,0,0)$.