Proof that $l^p$ with $1 \leq p < \infty$ is dense in $c_0$

Question

Proof that $l^p$ with $1 \leq p < \infty$ is dense in $c_0$ My definition of $c_0$ is the following:

$$ c_0 = \left\{ x \vert \lim_{k\to \infty} x_k = 0\right\}.$$

I want to prove it like this: every element of $c_0$ can be written as limit of $l^p$. First question, should I use the $p$ norm or the sup norm? Second question, how should I start given that I think this statement is true?

Answer 1

The space $c_0$ is usually considered as a subspace of $\ell^\infty$, and hence automatically inherits the sup-norm. For any given $x = (x_1, x_2, \ldots) \in c_0$, you can truncate it to obtain an $\ell^p$ sequence, i.e. you can consider $$ \hat x^{(n)} = (x_1, x_2, \ldots, x_n , 0, 0 ,\ldots) \in \ell^p. $$ Then $$ \Vert \hat x^{(n)} - x\Vert_{\infty} = \sup_{j \geq n+1} |x_j|, $$ which can evidently be made arbitrarily small by taking $n$ to be large, since $x_j \to 0$ as $j \to \infty$.