How to show a subspace is closed in $\ell^2$

Question

Let $F$ be the real sequence which satisfies $ \sup _{n} {|nx_n|}<\infty$ in $\ell^2$. Show that

1. $F$ are linear subspace of $\ell^2$.
2. $F$ is closed in $\ell^2$.

My idea:

1. is trivial.
2. As is known to all, $\ell^2$ is a Banach space. The subspace of a Banach space is closed if and only if it is complete (Is this right?).

So we only to prove $F$ is complete subspace. But how can I show that every Cauchy sequence converge in $F$? Or we need to show that the closure of $F$ is itself? Could someone give me some details. Thank you!

Answer 1

The set $F$ is not closed. To see it consider a sequence.

$$x^k =\left(\frac{1}{1} , \frac{1}{2^{\frac{3}{4} }}, \frac{1}{3^{\frac{3}{4} }} , ...,\frac{1}{k^{\frac{3}{4} }} , 0,0,0,....\right)$$

Clearly $$\sup_n |nx^k_n |<\infty$$ for all $k\in\mathbb{N} .$ Hence $x^k\in F.$ Moreover $x^k \to x$ in $\ell^2 $ where $$x= (n^{-\frac{3}{4}})_{n\in\mathbb{N}}\in\ell^2.$$ But $x\notin F.$