I'm stuck On the theoremTheorem 4.1.2, from C*-Algebras and Operator Theory by Gerald J. Murphy
4.1.2. Theorem. Suppose that $(p_\lambda)_{\lambda\in\Lambda}$ is a net of projections on a Hilbert space $H$.
(1) If $(p_\lambda)$ is increasing, then it is strongly convergent to the projection of $H$ onto the closed vector subspace $(\cup_{\lambda}p_\lambda(H))^-$.
(2) If $(p_\lambda)$ is decreasing, then it is strongly convergent to the projection of $H$ onto $\cap_\lambda p_\lambda(H)$.
I have no idea how to deal with. Any hints or suggestions would be welcome!
As established earlier in Murphy, an increasing net of projections that is bounded above by a self-adjoint element converges strongly to its supremum. Hence, for $(1)$ we must check that if $p$ is the projection onto $K= \overline{\bigcup_\lambda p_\lambda H}$, then $p$ is the supremum of the net $\{p_\lambda\}$.
This is rather easy, since $p \geq p_\lambda$ for all $\lambda$ because $pH \supseteq p_\lambda H$ by definition of $p$. Thus $p$ is an upper bound for this net.
Moreover, if $q \geq p_\lambda$ for all $\lambda$, then clearly $q \geq p$ and $$(p-q)p_\lambda = pp_\lambda-q p_\lambda = p_\lambda- p_\lambda = 0$$ so that $p-q$ annihilates the subspace $K$. Hence, $$0 = (p-q)p = p-qp= p-q$$ and thus $q=p$. Thus, $p$ is the least upperbound. Can you now do $(2)$ yourself?