Proving the Squeeze Theorem using the Epsilon Delta Definition for the Limit

Out of the many techniques there are for solving limits, the squeeze theorem is a fairly famous theorem that has the ability to evaluate certain limits by comparing with other functions. For those who do not know the squeeze theorem, it states the following:

Let A be some domain containing the point c, and let f, g, and h be defined on this common domain (except possibly at c). Suppose that for every x in A, f(x) ≤ g(x) ≤ h(x). Then, if lim(x → c) f(x) = lim(x → c) h(x) = L, it follows that lim(x → c) g(x) = L.

In this post, I will be going through a simple proof of this theorem using the epsilon delta definition for limits, and will finish with a simple application of this theorem.

Proving the Squeeze Theorem

To prove the squeeze theorem, I will be using the epsilon delta definition for limits which you can read more about in this post.

To begin, let f, g, and h be defined on A (except possibly at point c) and suppose that for every x in A, f(x) ≤ g(x) ≤ h(x). Additionally, suppose that lim(x → c) f(x) = lim(x → c) h(x) = L. Then, by the epsilon delta definition for the limit, we know the following statements are true:

Now, what we wish to show is the following:

To do this, we can start with |g(x) - L| and observe that we can rewrite this in terms of |f(x) - L| and |h(x) - L| in the following way using the…