How to show map is non-singular

Let $f:;mathbb{R}^ntomathbb{R}^n$ be differentiable. Suppose that
for all $xinmathbb{R}^n:$ $$lVert
mathrm{D}f(x)-mathrm{I}rVertleq frac{1}{2}$$ where
$lVertcdotrVert$ is the operator norm. I need to show $f$ must be a
diffeomorphism.

By using the contraction mapping theorem I have shown that $f$ is surjective, and also I have shown that $lVert f(x)-f(y)rVertgeqfrac{1}{2}lVert x -yrVert$ so $f$ must be injective.

I’d like to use the inverse function theorem, but to do that I need to show that $mathrm{D}f$ is non-singular. And this is where I’m stuck. Help?