My favorite theorem is the Squeeze Theorem:
Let I be an interval containing the point a. Let f, g, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have:
- $ g(x) \leq f(x) \leq h(x) $
- $ \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L. $
I have seen it used quite a bit in proving other theorems. For instance, in MA301 we used it when proving an example like:
2.1412.....
+1.3376.....
is 3.47 rounded to two decimal places, regardless of what the complete decimal expansion is.
