Norms are not equivalent in $c_0$

Question

Consider two norms in the space $c_0$: $$\lVert x \rVert = \sup \lvert x_i \rvert$$ and $$\lVert x \rVert _0=\sum 2^{-i} \lvert x_i \rvert$$ Prove that above two norms are not equivalent.

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I know $c_0$ with norm $\lVert \cdot \rVert$ is Banach space. And I guess that $c_0$ with the latter norm is not Banach. However, I can't find any example to show that.

Answer 1

Think about sequence $e_n \in c_0$, such that $e_n$ has $1$ on $n^{th}$ place and zero otherwise. Then $\Vert e_n \Vert_0 \rightarrow 0$, but $\Vert e_n \Vert = 1$ for all $n \in \mathbb N$.

Answer 2

It is clear that $\|x\|_0\le\|x\|$. To show that they are not equivalent you need to show that $\|x\|/\|x\|_0$ is unbounded. For this, try to choose sequences $a_n\in c_0$ such that $\|a_n\|/\|a_n\|_0\to\infty$ as $n\to\infty$. There is a pretty obvious choice.