I just came across a simple Functional Analysis exercise which I just dont know if I correctly solved. The problem states:
Consider the space $C(\mathbb{R}):=\{f: \mathbb{R} \longrightarrow \mathbb{C}; f$ continuous$\}$ equipped with the seminorm:
$$ p_{j}(f)=\sup \{|f(x)| ;|x| \leq j\}, \quad j \in \mathbb{N} $$
Prove that:
1. Let $f_{n} \in C(\mathbb{R}), n \in \mathbb{N}$, be defined by $f_{n}(x)=x-n+|x-n|, x \in \mathbb{R}$. Show that $\lim _{n \rightarrow \infty} f_{n}=0$.
2. Show that $C(\mathbb{R})$ is a complete metric space.
Regarding Completeness
Let $\left(f_{n}\right)_{n}$ be a Cauchy sequence in $C(\mathbb{R})$. Now, noting that:
$$ H=\left\{\left|f_{n}(x)-f_{m}(x)\right| ;|x| \leqslant j\right\} \subseteq G \doteq\left\{\left|f_{n}(x)-f_{m}(x)\right|x \in \mathbb{R}\right\} $$
We conclude that $\sup H \leq \sup G$. Noting that $\sup G$ is precisely the $\sup$ norm, means we have: $$ 0 \leq p_{j}\left(f_{n}-f_{m}\right) \leqslant\left\|f_{n}-f_{m}\right\|_{\infty} $$
Since $C(\mathbb{R})$ is complete with respect to the sup norm, we have that our initial Cauchy sequence $\left(f_{n}\right)_{n}$ converges to a function $f$, which should be continuous as well, with respect to the $\sup$ norm. This implies that there exists $n_{0}$ such that for $n, m > n_{0}$, we have that $\left\|f_{n}-f_{m}\right\|_{\infty} < \epsilon$, which implies:
$$ 0 \leq p_{j}\left(f_{n}-f_{m}\right) \leqslant\left\|f_{n}-f_{m}\right\|_{\infty} < \epsilon $$
Consequently, $p_{j}\left(f_{n}-f_{m}\right)$ is a Cauchy sequence which converges to $f$ the same continuous function, turning our initial space equipped with the seminorm $p_{j}$ complete.
Regarding the limit $\lim _{n \rightarrow \infty} f_{n}=0$:
We want to show that, given $\epsilon > 0$, there exists $n_{0}$ such that for $n > n_{0} \implies p_{j}(f_{n}, 0) < \epsilon$. Since $\mathbb{N}$ is unbounded, there exists $N$ such that for every $n > N$, the difference $|x-n| = n-x$. This implies that:
$$ \begin{aligned} p_{j}(f_{n}, 0) & \doteq \sup \{|x-n+|x-n|| ;|x| \leqslant j\}=\\ &=\sup \{|0| ;|x| \leqslant j \mid=0<\varepsilon \end{aligned} $$
which proves that $\lim _{n \rightarrow \infty} f_{n}=0$ with respect to the seminorm $p_{j}$.
Questions:
- Are the arguments ok for each of the questions? If no, what is missing? Is it well written?
Thanks in advance, Lucas