Square root is operator monotone This is a fact I've used a lot, but how would one actually prove this statement?
Paraphrased: given two positive operators $X, Y \geq 0$, how can you show that $X^2 \leq Y^2 \Rightarrow X \leq Y$ (or that $X \leq Y \Rightarrow \sqrt X \leq \sqrt Y$, but I feel like the first version would be easier to work with)?
Note: $X$ and $Y$ don't have to commute, so $X^2 - Y^2$ is not necessarily $(X+Y)(X-Y)$.

Define $Y = X + k$, where $k$ is $\ge 0$. Then $Y^2 = X^2 + 2kX + k^2$
