I'm solving Problem II.3.7 from textbook Analysis I by Amann/Escher
where $\mathbb K$ denotes $\mathbb R$ or $\mathbb C$, and $\ell_\infty$ is defined as
and $c_0$ is defined as
Could you please verify whether my attempt is fine or contains logical gaps/errors? Any suggestion is greatly appreciated!
My attempt:
Assume that $(x_n)$ is a sequence in $c_0$ that converges to $x \in \ell_\infty$. Then each $x_n$ is a sequence in $\mathbb K$ that converges to $0 \in \mathbb K$. To prove that $c_0$ is closed, we show that $x \in c_0$, i.e. $x$ is a sequence in $\mathbb K$ that converges to $0 \in \mathbb K$.
Because $(x_n) \to x$, $\Vert x_n - x \Vert_\infty < \varepsilon/2$ for all $n > N$ and thus $\sup_{m \in \mathbb N} |x_n^m - x^m| < \varepsilon/2$ for all $n > N$. Here $x_n^m,x^m$ are the $m$-th elements of the sequences $x_n$ and $x$ respectively. We then take $n_0 > N$. Since $x_{n_0}$ is a sequence in $\mathbb K$ that converges to $0 \in \mathbb K$, there exists $M \in \mathbb N$ such that $|x_{n_0}^m| < \varepsilon/2$ for all $m > M$.
We have $|x^m| \le |x_{n_0}^m - x^m| + |x_{n_0}^m| \le \sup_{m \in \mathbb N} |x_{n_0}^m - x^m| + |x_{n_0}^m| < \varepsilon/2 + \varepsilon/2 = \varepsilon$ for all $m > M$. As such, the sequence converges to $0 \in \mathbb K$. This completes the proof.