Hello, could you please help me to solve this problem:

Let $\displaystyle B(H)$ - the algebra of bounded linear operators on a Hilbert space $\displaystyle H$, and $\displaystyle x\in B(H)$ is positive. Then sequence $\displaystyle x (\frac{1}{n}+x)^{-1}$ is monotone increasing to the range projection of $\displaystyle x:\quad [x].$