Let $a$ be a positive element in a $C^*$ algebra $A\subseteq B(H)$.
Consider the S.O.T-lim of the sequence $(x+\frac{1}{n}1_{H})^{-1}x$. The limit exists in $B(H)$ since the sequence is monotone increasing bounded sequence of positive elements.
I want to show that the limit is equal to $1_H$.
I've tried the Dini's uniform convergence theorem (with functional calculus in mind), but the sequence does not converge pointwise to a continuous function. Actually, the convergence is to $0$ at $0$ and $1$ at $(0,\infty)$ on $\sigma(x)\subseteq [0,\infty)$.
Then, I've tried to use Borel functional calculus. However, I'm not sure that I can "forget" the $0$ at zero since it is a set of measure $0$ and claim that it converges to $1$.
Thanks for any help.