Is the Product of Banach Spaces a Banach Space?

Question

Let $X$ and $Y$ be two Banach spaces (not necessarily possessing the same norm). The product space $X×Y=Z$ is given the max norm, i.e. $\max(\Vert x\Vert, \Vert y\Vert)$, where $x$ is given the norm of $X$ and $y$ is given the norm of $Y$, and where $(x,y)$ is an element of $Z$. Prove that the product space $Z$ is Banach.

Answer 1

$z_n=(x_n,y_n)$ is a cauchy (convergent) sequence in $Z$ iff $(x_n)$ and $(y_n)$ both are cauchy (convergent).