# 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.