The book in question is "C* algebras and their automorphism groups" from Petersen et al. The question is simple and I imagine it should be easy, as they do not say much about that, but I am having difficulties proving that the sequence, for $x\geq 0$, $(1/n+x)^{-1}x$ converges (I suppose they mean strongly) to $[x]$, where $[x]$ is the range projection (it is defined in the picture). I can see that the sequence is monotonically increasing and bounded, so it should converge strongly to an operator $y\in B(H)$, but why $[x]$?
The functions $f_n:[0, \infty )\to {\mathbb R}$ defined by $$ f_n(t)=(1/n+t)^{-1}t $$ are bounded by 1 and converge pointwise to the characteristic function $1_{(0,\infty )}$. Therefore, for every positive operator $x$, one has by [1, Lemma (3.5.5)] that $f_n(x)$ converges strongly to $1_{(0,\infty )}(x)$, which turns out to be the range projection of $x$.
[1] Sunder, V. S., Functional analysis: spectral theory, Birkhäuser Advanced Texts. Basel: Birkhäuser. ix, 241 p. (1997). ZBL0919.46002.