Proving that $c_c(\mathbb N)$ is a dense subset of $l^p(\mathbb N)$

Question

Proving that $c_c(\mathbb N)$ is a dense subset of $l^p(\mathbb N)$

$c_c(\mathbb N)$-space of sequences which are zero after finitely many terms. $$l_p=\{(x_i)^{\infty}_{i=1}|\sum_{i=1}^{\infty}|x_i|^p<\infty\}$$

Answer 1

Let $e_n$ the sequence which is only $0$ except in the $n^{th}$ term where it is $1$. Let $f \in l^p$ and let $u_n = \sum_{k=1}^n f(k)e_k$. We will show that $u_n \in c_c$ converges to $f$ in $l^p$, so it shows that $c_c$ is dense in $l^p$.

$$||f-u_n||_p=||f-\sum_{k=1}^n f(k)e_k||_p=\sum_{k=n+1}^{\infty}|f(k)|^p$$

Which tends to $0$ when $n$ tends to $\infty$. So $c_c$ is dense in $l^p$.