I would like to prove the following proposition:
Let $\tau$ be a topology. A finite intersection of elements of $\tau$ is also in $\tau$.
The proof is by induction on the number of elements in the intersection.
Base case: an element of $\tau$ is in $\tau$ by definition.
Suppose that the statement holds for intersections of $k\lt n$ elements. Let $S$ be an intersection of $n$ elements. The intersection of the first $n-1$ elements is in $\tau$ (by the induction hypothesis). Now we have an intersection of two elements of $\tau$ which is an element of $\tau$ (again by the induction hypothesis).
Is my proof correct?