The image you uploaded contains mathematical problems. Let's go over them:

1. The first part involves projections in an inner product space $X$. It asks to show that if $P$ and $Q$ are two projections, then $R(P) \subseteq R(Q)$ implies $P = Q$.

2. The second part concerns a Hilbert space $H$, with a non-empty convex subset $M$. It asks to prove that for a sequence $(x_n)$ in $M$, where $\| x_n \|$ approaches a value $\alpha$, and $\alpha = \inf \| x \|$, we need to prove a certain result.

Would you like help solving both of these problems? Feel free to indicate if you'd prefer to start with one or the other!