If I add a positive definite matrix, do I increase the norm?

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I want to show that if $\|A' - B'\|\geq \|A - B\|$ as soon as $A' \succeq A$ and $B' \preceq B$. Is this true?

I could show it if I could show that $\|C + D\| \geq \|C\|$ when $D$ is positive definite.