I was reading a book on compressive sensing wherein they mentioned that in the limit, the zeroth norm of a vector is the number of non-zero elements in that vector. That is,
$$\lim_{p\rightarrow 0} \Vert x\Vert _p = \Vert x\Vert_0 = |\mathrm{supp}(x)|$$
where $\mathrm{supp}(x)$ is the number of non-zero elements in $x$.
I could not prove this. Any help would be greatly appreciated.
This is not true in this form. If $k>0$ out of the $n$ components of $x$ are nonzero, let $c=\min\{\,|x_i|:1\le i\le n, x_i\ne 0\,\}$. Then $c>0$. Also let $C=\max\{\,|x_i|:1\le i\le n\,\}$. Then $$k^{1/p}c=(kc^p)^{1/p}\le\|x\|_p=\left(\sum |x_i|^p\right)^{1/p}\le (kC^p)^{1/p}=k^{1/p}C $$ As $p\to 0$, both bounds tend to $+\infty$ if $k\ge2$ and for $k=1$ they reamin constant at $c$ and $C$, that is unrelated to $k$.
However, we see that $$kc^p\le \|x\|_p^p\le kC^p $$ and here both bounds tend to $k$ as $p\to 0$.
