The set of elementary tensors is weakly closed in projective tensor products

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.