  In this paper, we show that $\infty$-adic multiple zeta values associated to
the function field of an algebraic curve of higher genus over a finite field
are not zero, under certain assumption on the gap sequence associated to the
rational point $\infty$ on the given curve. Using arguments and results of
Sheats and Thakur for the case of the projective line, we calculate the
absolute values of power sums in the series defining multiple zeta values, and
show that the calculation implies the non-vanishing result.
