Does convergence for increasing sequences of projections imply convergence for all sequences of projections?

Question

Let be $\{P_n\}_n$ an increasing sequence ($P_{n+1} \ge P_n$) of finite dimensional projection operators on a separable Hilbert space $H$. Let $f:H \to \mathbb{C}$. Let us fix $f_n(x):=f(P_nx)$. If $f_n$ converges to $\tilde{f}$ for any increasing sequence such that $P_n \to I$ strongly (with $I$ identity operator), is it true that $f_n$ converges to $\tilde f$ for any sequence $\{ P_n\}_n$ such that $P_n \to I$ strongly?