I have been reading Felix E. Browder's Convergence Theorems for Sequence of Nonlinear Operators in Banach Space and I was hoping I could find answers to a couple of questions I have about the paper. Consider Lemma 6.

Let $F$ be a closed convex subspace of a Hilbert space H. Let $\{u_n\}$ be a sequence in $H$ such that

- For each $f\in F$, $\|u_n-f\|$ is a non-increasing sequence
- Each weak limit point of the sequence $\{u_n\}$ lies in $F$.
Then $u_n \to f_0$ weakly for some point $f_0$ in $F$.

I can follow most of the proof. However, is the bound $\|u_n\|\leq \|u_1\|+\|f\|$ for a fixed $f$ necessarily correct? It seems that it should be $\|u_n\|\leq \|u_1\|+2\|f\|$. In any case, I am not sure how one obtains $p(f):=\lim_{n\to \infty} \|u_n-f\| \ge \|f\|-M_0$ (I don't think he mentions what $M_0$ stands for, but I strongly suspect it is the bound of $\|u_{n}\|$, for which the inequality would make sense).

Also, how does lower semi-continuous $p$ assumes minimum at some point $f_0$ of $F$? Is there is a version of the following theorem (link) for closed convex subsets of Hilbert spaces and lower semi-continuous $p$?

Further, consider Lemma 7.

Let $X$ be a strictly convex Banach space. $U$ is a non-expansive mapping of a convex subset $C$ of $X$ into $X$. Then the fixed point set of $U$ in $C$ is convex.

The proof starts off by considering two fixed points $f_0$ and $f_1$ of $U$ and trying to prove that $y_t= (1-t)f_0+tf_1$ is a fixed point for $0\leq t \leq 1$. I have been able to follow the proof except that I don't see how the remark that $U(y_t)$ lies on the line segment between $f_0$ and $f_1$ follows. Thanks!