Embeddings of $n$-dimensional separable metric spaces into the product of Sierpinski curves

We give a short proof of the following fact: the set of embeddings of any $n$-dimensional separable metric space $X$ into a certain $n$-dimensional subset of the $(n+1)$-product of Sierpinski curves $\Sigma$ is residual in $C(X, \Sigma ^{n+1})$.