Brouwer Fixed Point Theorem implies Invariance of Domain Theorem

Is there a simple and elementary proof that Brouwer's Fixed-Point Theorem implies Invariance of Domain Theorem? By 'simple and elementary' I mean proof that:

• Do not use tools such as degree theory or a sophisticated topological invariant.

• be as self contained as possible.

I have tried to adapt the proof that Banach's Fixed-point Theorem implies the Inverse Function Theorem but did not succeed.

Thank you

Brouwer Fixed Point Theorem. Let $f:U\to\mathbb{R}^n$ be a continuous function defined in an open set $U\subset\mathbb{R}^n$. Let $K\subset U$ convex and compact. If $f(K)\subset K$ then there is $x_0\in K$ such that $f(x_0)=x_0$.

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Invariance of Domain Theorem. Let $f:U\to\mathbb{R}^n$ be a continuous function defined in an open set $U\subset\mathbb{R}^n$. If $f$ is one to one function then $f(U)$ is a open subset of $\mathbb{R}^n$.

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• Do you have a reason (other than the fact that Brouwer proved both theorems) to expect that such a proof exists? – Andreas Blass Apr 5 '18 at 13:26