If $f:\mathbb R^n \to\mathbb R^n$ is continuous and such that for each $x$ in the domain $\|f(x)-x\|\leq1$ holds, how can I prove that then $f$ must be onto?
As a hint, I have been told, for an $a\in\mathbb R^n$ look for a fixed point of the function
$$g_a(x):=x+a-f(x+a)$$
2nd hint: Suppose that the value $a$ is not attained.
What can you say about fixed points of $g_a$ ?
And what does Brouwer give you when you look at the map of the closed unit ball under $g_a$?