Triangle inequality with spectral norm Let $A$ be a real square matrix, I have to prove that
$$\left \| A \right \|=\sqrt{\lambda_{\text{max}}(A^*A)}$$
defines a norm. I don't know how to prove the triangle inequality. I have already proved that $\|A\|=\|A\|_2=\sup_{\|x\|_2=1} \|Ax\|_2$, but the exercise is to prove without using it. 

Preliminary answer: This inequality follows from Theorem 3.3.16 in Horn and Johnson (1994) - Topics in Matrix Analysis, and I think they don't use the equality you said in their proof.
Although I don't understand their proof, I am trying to do so since I have this question too (from research). I will update this answer later, once I understand the proof.
