I want to prove the following fact without using topological degree theory or related algebraic topology

Let $h:\overline{B}(0,1)\to \mathbb{R}^n$ be a continuous map such that $|h(x)-x|\leq \delta$ when $|x|=1$. Then $B(0,1-\delta)\subset h(B(0,1))$.

The result is obvious if one uses topological degree theory, since $h$ is homotopic to the indentity map when restricted to $B(0,1-\delta)$. Or with more effort, one can use Brouwer fixed point theorem to prove it. Here I want an elementary proof of this fact, which does not involve any algebraic topological concept.