Denoting $c_0 = \{(x_n)_{n=1}^{\infty} : \lim_{n \to \infty} x_n = 0 \}$, I want to prove that the normed space $(c_0, \| \cdot \|)$ is not a Banach space, where $\|x\|= \sum\limits_{n=1}^{\infty}\frac{|x_n|}{2^n}$ .
I know that I need to find a Cauchy sequence in $c_0$ that converges to an element that does not belong in $c_0$, but I'm stuck.
I've tried using the sequence $(n/k)_{k=1}^{\infty} \forall n \in \mathbb{N}$, which is a Cauchy sequence, but I don't think it works.
Any help would be greatly appreciated.
Proving that $c_0$ with the norm $\|x\|= \sum\limits_{n=1}^{\infty}\frac{|x_n|}{2^n}$ is not Banach
functional-analysisreal-analysis
Best Answer
Consider the sequence $X_{k,n}$ defined as $1$ for $n\leq k$ and $0$ for $n>k$. Then obviously $X_{k,n}\in c_0$ for every $k\in\mathbb N$. In addition $X_{k,n}\stackrel{||\cdot||}{\longrightarrow}X_{\infty,n}$ defined as $1$ for every $n\in\mathbb N$. However $X_{\infty,n}\not\in c_0$.