TITLE: An exercise of positive element in C*-algebra
QUESTION [5 upvotes]: Let $A$ be a unital C*-algebra and $\{b_{n}\}$ be a positive invertible sequence in $A$. If $||1_{A}-b_{n}||\rightarrow 0$, can we conclude $||1_{A}-b_{n}^{-\frac{1}{2}}||\rightarrow 0$ ?

REPLY [4 votes]: Since $b_n\geq0$ and $b_n\to1$, eventually we will have $\sigma(b_n)\subset (k_1,k_2)$ for $k_1,k_2> 0$ and sufficiently close to $1$. Then
$$
\|1_A-b_n^{-1/2}\|=\|b_n^{-1}(1_A-b_n)(1_A+b_n^{-1/2})^{-1}\|\\
\leq\|b_n^{-1}\|\,\|1_A-b_n\|\,\|(1_A+b_n^{-1/2})^{-1}\|
\leq k_1^{-1}\,(1+k_2^{-1/2})^{-1}\,\|1_A-b_n\|$$