Weakly sequential completeness of the projective tensor product $L^{\lowercase{p}}[0,1]\hat{\otimes}X, 1 < \lowercase{p} < \infty$

D. R. Lewis (1977) proved that for a Banach space $X$ and $1 < p < \infty$, $L^p[0,1]\hat{\otimes}X$, the projective tensor product of $L^p[0,1]$ and $X$, is weakly sequentially complete whenever $X$ is weakly sequentially complete. In this note, we give a short proof of Lewis's result, based on our sequential representation (2001) of $L^p[0,1]\hat{\otimes}X$.