Limit to zero of the $p$-norm I have the $p$-norm defined as $$\|x\|_p=\left(\sum_{k=1}^n|x_k|^p\right)^\frac{1}{p}.$$  I am trying to find the limit as $p\to0^+$ of $\|x\|_p$.  I've seen it defined as $\{x_k:x_k\neq0\}$. Why is this true?  Is there a way to prove this?

Let $m=\min\{|x_i|:\ x_i\neq0\}$, and $k$ the number of non-zero $x_i$. Then 
$$||x||_p=m\cdot\left(\sum_{i=1}^{n}\left|\frac{x_i}{m}\right|^{p}\right)^{1/p}\to\begin{cases}0&\text{ if }k=0\\m&\text{ if }k=1\\\infty&\text{ if }k>1\end{cases}$$
The reason for the value when $k>1$ is that the $\sum$ above has $k$ terms $\geq1$, and $1/p\to\infty$.
