$P_n T P_nh$ tends to $Th$ as $n$ tends to infinity in Hilbert space

Question

Let $H$ be a Hilbert space and $P_n$ be the orthogonal projection onto an orthonormal basis of $H$. If $T$ be a bounded linear operator on $H$ , then does $P_nTP_nh\to Th$ as $n$ tends to infinity.

I think yes, by the completeness of hilbert space. But, is there a rigorous proof of the assertion?Thanks beforehand.