Understanding the proof that monotone operators are locally bounded

Question

I am trying to understand the proof that monotone operators are locally bounded:

The question I have is why is the inequality highlighted yellow true? The problem I have is that the inequality of 5.2 is not necessarily preserved under taking norms as the left-hand side may have a larger modulus (norm) that the right-hand side. If that wasn't the case, everything would be fine.

Answer 1

You are right, this proof is not correct. To obtain the claim, repeat estimate (5.2) with $y$ replaced by $-y$. And the claimed inequality (final inequality in yellow part) should be true with $\|A(x_0+y)\|$ replaced by $\|A(x_0+y)\|+\|A(x_0-y)\|$.