To show this, we need:

Fortunately we can use a trick involving the Cauchy-Schwartz Inequality to prove this. Otherwise, it would be very messy to see how making the term with the square root small will result in making the first term small. The points of this example are twofold: The nice trick with Cauchy-Schwartz is a good application of that theorem. On the other hand, we really can't expect something like that to come to our rescue in even slightly more complicated situations. We need to have tools for proving continuity of functions in Euclidean space, because the estimates involving the metric can become very messy.