Infinite Intersections Of Open Sets

Let $U_{\alpha}$ $(\alpha \in A)$ be a collection of open sets in ${\mathbb R}^2.$ If $A$ is finite, then the intersection $U = \bigcap\limits_\alpha U_{\alpha}$ is also an open set. Here is a proof:

Suppose $x\in U.$ For each $\alpha \in A,$ let $B_{\alpha}$ be a ball of some positive radius around $x$ which is contained entirely inside $U_{\alpha}.$ Then the intersection of the $B_{\alpha}$ is a ball $B$ around $x$ which is contained entirely inside the intersection, so the intersection is open.

$($Here a ball around $x$ is a set $B(x,r)$ ($r$ a positive real number) consisting of all points $y$ such that $|x-y|<r.$ In ${\mathbb R}^2$ it is an open disk centered at $x$ of radius $r.)$

Where does this proof go wrong when $A$ is infinite?