How can I prove the Initial Value Theorem?

Question

I want to prove that, for a discrete causal signal $x[n]$, $$x[0] = \lim_{z\to \infty} X(z)$$ where $X(z)$ is the $z$-transform of $x[n]$.

Answer 1

A causal signal is one that $x[n]=f(n)u[n]$ or $x[n]=0,\forall n<0$.

Looking at the definition of $z$-transform:

$$X(z)=\sum_{n=0}^{\infty}x[n]z^{-n}=\sum_{n=0}^{\infty}x[n]\frac{1}{z^{n}}$$

when $z\to\infty$, the term $\frac{1}{z^{n}}$ vanishes all the terms of the series, except if $n=0$ where it is just one. So only $x[0]$ remains from the series.