Show that the linear functional is unbounded in $C_{00}$. defined as $T$ is defined as $T(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}},$

Given a linear functional $T: C_{00}\to C_{00}$. Where $C_{00}$ is space sequences with finitely many non-zero terms with $\ell_2$ norm.

$T$ is defined as $$T(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}},$$Show that this functional is unbounded.

Initially Idea was to choose $x_n=\{1\}_{1\leq k\leq n}\in C_{00}$This gives $$T(x_n)=\sum_{k=1}^{n} \frac{1}{\sqrt{k}},$$

This gives $T(x_n)$ diverges hence functional is bounded.

Is this a correct approach.