I am trying to solve problems. Two problem is as follows.
Q1 suppose $f_k$ defined on $\Bbb R^n$ converge uniformly to a function $f$. suppose that each $f_k$ is bounded, say by $A_k$, prove that $f$ is bounded.
here is my proof
given $\varepsilon =1 $ , $\exists N \in \Bbb N$ such that $$||f_k(x)-f(x)||\lt 1 $$ for all $x \in \Bbb R^n$ and $k \ge N$
since $f_k$ is bounded by $A_k$, by triangle inequality $$||f(x)||\le ||f_k(x)-f(x)||+||f_k(x)|| \le 1+A_k$$
if $1\le k \le N-1$, Let $M_0=max\{A_1+1,A_2+2,\cdots, A_{N-1}+1 \}$
Then, for all $k=1,2,\cdots N, N+1, \cdots $ $$||f(x)|| \le M = max\{M_0,A_k+1\}$$
Thus $f$ is bounded
Q2 Let V be a complete normed vector space.
(a) Let $(f_n)$be a cauchy sequence in $C([a,b],V)$. show that for each $x\in[a,b]$, $(f_n(x))$is cauchy, and so define the pointwise limit $f(x) =\lim_{n\to >\infty}{f_n(x)}$
(b) prove that $(f_n)$ converges uniformly.
(c) prove that f is continuous, and deduce that $C([a,b],V)$ is complete.
I was stucked by problem above Q2, because no norm isn't given in $C([a,b],V)$. I wander if it is fault of book. or can I Prove above problem for arbitrary norm in $C([a,b],V)$? if the norm is sup norm in $C([a,b],V)$, I might be able to solve this problem. please advise me ! I need your help. thank you