The following is a part of a proof in Takasaki's Operator theory:
Let $\epsilon>0$ and $A$ is a C*-algebra. For an $x\in A$, put $h=x^*x$ and $u_\epsilon=(h+\epsilon)^{-1}h$. We have then $$||x(1-u_\epsilon)^\frac{1}{2}||=||\epsilon x(h+\epsilon)^\frac{-1}{2}||$$$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\epsilon|| (h+\epsilon)^\frac{-1}{2}x^*x(h+\epsilon)^\frac{-1}{2}||^\frac{1}{2}$$$$~~~~~~~~~~~~~~~~~~~~~~~=\epsilon||(h+\epsilon)^{-1}h||\leq \epsilon$$
I do not know how he concludes the last equation. I define $u_\epsilon:=(h+\epsilon^2)^{-1}h$, and I think it follows like below,
$$||x(1-u_\epsilon)^\frac{1}{2}||=||\epsilon x(h+\epsilon^2)^\frac{-1}{2}||$$$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\epsilon|| (h+\epsilon^2)^\frac{-1}{2}x^*x(h+\epsilon^2)^\frac{-1}{2}||^\frac{1}{2}$$$$~~~~~~~~~~~~~~~~~~~~~~~=\epsilon||(h+\epsilon^2)^{-1}h||^\frac{1}{2}\leq \epsilon$$
Please check my attempt and help me to understand Takasaki's way. Thanks in advance.
I think it's a typo and my way is correct.