I'm a third year student in the math program and I really like to explore interesting results about math, sometimes I think that I focus on things that are not even within my reach. I recently found a book on Lipschitz functions where I found a theorem called Kirszbraun's theorem which states the following: let $A \subset \mathbb{R}^n$. If $f \colon A \rightarrow \mathbb{R}^m$ is a $L$-Lipschitz function, then there exists an extension $F \colon \mathbb{R}^n \rightarrow \mathbb{R}^m$ of $f$ which is also $L$-Lipschitz, that is, $F|_A =f$ and $L(F)=L(f)$.
Later they generalize this result for Hilbert spaces receiving the name of the Kirszbraun-Valentine theorem.
I had the audacity to start reading the proof in the case of Hilbert spaces, but there are really many things that I don't understand and that I haven't seen in the program yet. I studied many different proofs and some authors in their proofs use polytopes, others use filters, others use Kirszbraun Property (K), etc. I'm not really familiar with these concepts. Therefore, I would like to know if there is a more affordable proof for me, at least for the case in Euclidean spaces (I think that in this case the proof should be a little less rigorous). I would appreciate very much if someone can help me to get a more understandable proof for me.
I'm completely familiar with Lipschitz functions and it is very surprising to me that this type of extension is possible.