# Find the completion of the metric space.

Let $$\ell^1(\mathbb{N})=$$ $$\{(a_n): a_n ∈ \mathbb{R} , \sum |a_n| \lt \infty \}$$, and define the distance function $$d, d_* : \ell^1(\mathbb{N})\times\ell^1(\mathbb{N}) → [0, \infty)$$ by

$$d(a,b)=\sum |a_n - b_n|$$, $$d_*(a,b)= sup \,|a_n - b_n|, a, b ∈ \ell^1(\mathbb{N})$$.

I'd like to find the completion of the metric space $$(\ell^1(\mathbb{N}), d_*)$$.

I don't know how to start...

Could you give me a hint or any help will be appreciated.

## 1 Answer

The completion is $$c_0$$, the space of sequences with limit $$0$$. Note that if $$a_n \to 0$$ then $$(a_1,a_2,...,a_k,0,0,...)$$ is a sequence in $$\ell^{1}$$ converging in the sup metric to $$(a_n)$$. Conversely If you can approximate $$(a_n)$$ under $$d_{*}$$ by an element of $$\ell^{1}$$ then $$a_n \to 0$$ because every sequence in $$\ell^{1}$$ has limit $$0$$.