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

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?

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 $$j \to \infty$$.