TITLE: proving that a function from $\Bbb R^2$ to $\Bbb R^2$ is surjective
QUESTION [7 upvotes]: Let $f: \Bbb R^2 \to \Bbb R^2$ be a continuous function such that $|f(x) - x| < 2017 \space$ $\forall$ $x\in \Bbb R^2$.
prove that $f$ is surjective.   
please help me prove this statement.
thank you very much.

REPLY [5 votes]: If $\|x\|\to\infty$ then clearly $\|f(x)\|\to\infty$. This means that after a $1$-point compactification of $R^2$, you can continuously extent $f$ to a function $S^2\to S^2$ via $f(\infty)=\infty$. Further, $f: S^2\to S^2$ is homotopic to the identity map, because the homotopy $(x,t)\mapsto x+t(f(x)-x)$ can be continuously extended to a pointed homotopy of the sphere via $(\infty,t)\mapsto\infty$. Hence the degree of $f$ is $1$ and $f$ is surjective.