  In this short note, we prove that the set of elementary tensors is weakly
closed in the projective tensor product of two Banach spaces. As a result, we
are able to answer a question from the literature proving that if $(x_n)
\subset X$ and $(y_n) \subset Y$ are two weakly null sequences such that $(x_n
\otimes y_n)$ converges weakly in $X \widehat{\otimes}_\pi Y$, then $(x_n
\otimes y_n)$ is also weakly null.
