Show that $|\Gamma| = \lim_{\sigma \to 0}{\sum_{k=0}^{n-1}{|\mu(t_{k+1})-\mu(t_k)|}}$ for $\Gamma$ curve on complex plane

Question

Show that $|\Gamma| = \lim_{\sigma \to 0}{\sum_{k=0}^{n-1}{|\mu(t_{k+1})-\mu(t_k)|}}$ for $\Gamma$ curve on complex plane, where $\Gamma: \{\mu,[\alpha,\beta]\}, \sigma = max\{t_{k+1}-t_k\}$. By book has defined that $|\Gamma| = \sup\{\sum_{k=0}^{n-1}{|\mu(t_{k+1})-\mu(t_k)|}\}$. Also, it is obvious that $\lim_{\sigma \to 0}{\sum_{k=0}^{n-1}{|\mu(t_{k+1})-\mu(t_k)|}} \leq |\Gamma|$, because $|\Gamma|$ doesn't depend on $\sigma$. Yet I can't show that $|\Gamma| \leq \lim_{\sigma \to 0}{\sum_{k=0}^{n-1}{|\mu(t_{k+1})-\mu(t_k)|}}$