Norm inequality about semi-definite matrices

Question

If $A$, $B$ and $A-B$ are positive semi-definite, can we say that $\Vert A \Vert_2 \geq \Vert B \Vert_2$ where $\Vert\cdot\Vert_2$ indicates the spectral norm?

The answer to this question has mentioned that

... On the other hand, $0\leq X_1\leq X_1+X_2$ implies that $\|X_1\|\leq\|X_1+X_2\|$ ...

but how to prove it?

Answer 1

For a positive semidefinite matrix, $$\|A\|_2 = \sup_{\|x\|=1} \langle Ax, x\rangle$$ (reference). Since $A-B$ is positive semidefinite, $$\|A\|_2 = \sup_{\|x\|=1} \langle Bx + (A-B)x, x\rangle \ge \sup_{\|x\|=1} \langle Bx , x\rangle =\|B\|_2 $$