Let $(B,\vvvert\mathord{\cdot}\vvvert)$ be a normed algebra and let $\mathcal{B}$ be its separated completion. Let $A$ be a sub-$k$-algebra of $B$, equipped with the restriction algebra norm of $\vvvert\mathord{\cdot}\vvvert$, and let $\mathcal{A}$ be the separated completion of $(A,\vvvert\mathord{\cdot}\vvvert)$. Assume that $\mathcal{A}$ is an affinoid algebra. If $B$ is integral and is finite over $A$, then $\mathcal{B}$ is Banach finite over $\mathcal{A}$. Therefore $\mathcal{B}$ is an affinoid algebra.