I have seen in many compressed sensing books which say that
$$\lim_{p\to 0} \|x\|_p = \lim_{p\rightarrow 0}\left(\sum^n_{i=1}|x_i|^p\right)^{1/p} = \mbox{supp}\left(x\right)=\text{#} \{x_k:x_k \neq 0\} = \texttt{no. of non-zero elements of x}$$
where $\mbox{supp}\left(x\right)$, or support of $x$, is the number of non-zero elements of $x$.
I have no idea how to prove it. I have seen various other questions and their corresponding answers but none clearly states a process to obtain the proof. Is there any book where a proof is given? How should one start this proof?
As mentioned in a comment, the correct statement is that: $$ \lim_{p \rightarrow 0^+} \|x\|_p^p = |\text{supp}(x)| $$ Which you can derive from the simple observation: $$ \lim_{p \rightarrow 0^+} |x_k|^p = \begin{cases}0 & x_k = 0 \\ 1 & x_k \ne 0\end{cases} $$
I think you may have heard people say things like "minimizing the support of $x$ is like minimizing $\|x\|_p$ for small $p$". But keep in mind that if $f$ is a nonnegative function and $p$ is a positive number, then minimizing $f$ is equivalent to minimizing $f^p$, so the two notions are sometimes conflated.