My teacher gave this lemma in class without proof, I tried to prove it but found it seemed a little bit tricky:
Let a, b, c be real numbers. Then for any
$ \varepsilon\in(0, 1)$ we have $|\left| a-c\right|^{2}-|b-c|^{2}|\leq \frac{12}{\varepsilon} | a-b|^{2}+2 \varepsilon |a-c|^{2}$.
I found the original paper of this lemma(in page 39 of this paper, and it refers to this paper), but it uses very complicated tools. Is there any elementary proof?