The first step is to show that if then . This is easily seen by making the substitution in the integral . That shows that if is close to 1 and is close to , then is close to .
The next step is to show that if is continuous with compact support then is small if is close to 1. This is an easy consequence of the fact that is uniformly continuous.
Now you just have to put those two steps together and use the triangle inequality: . If is close enough to and is sufficiently close to 1, then all three terms on the right side of the inequality can be made arbitrarily small.