Can someone explain this solution to me (Johnson-Lindenstrauss, Triangle Inequality, Preserving Scalar Products)

Question

in my exam prep I had this solution (Exercise was to show the last line can be derived from the first line). The problem is that I don't get what happens between the "Thus". There was some hint about taking the norm, using triangle/cauchy inequality and the Johnson-Lindenstrauss-Lemma about preserving scalar products but I just don't get it :

Answer 1

Take the norm of both sides.

• Presumably you have a form of the Johnson-Lindenstrauss lemma that implies $$|\langle f(v_1) , f(v_2) \rangle | \le |\langle v_1, v_2 \rangle + \epsilon|.$$
• Presumably $\|v_2\|=1$.