The Squeeze Theorem is a useful tool in math that helps us understand how certain sequences behave.

To put it simply, the Squeeze Theorem says:

If we have a sequence called ( a_n ) that is “squeezed” between two other sequences named ( b_n ) and ( c_n ) like this:

$$b_n \leq a_n \leq c_n$$

for every number in a certain set, and if both ( b_n ) and ( c_n ) approach the same number ( L ) as we look at more and more terms (as ( n ) gets really big), then we can say that:

$$\lim_{n \to \infty} a_n = L.$$

An Example

Let’s see how this works with an example:

1. Define the Sequence: Let’s take the sequence ( a_n = \frac{n}{n+1} ).

2. Find Some Bounds: We can find two sequences to squeeze it. We can use ( b_n = \frac{n}{n+1} ) and ( c_n = \frac{n+1}{n+1} = 1 ).

3. Check the Limits:

   • For ( b_n ):
   $$\lim_{n \to \infty} b_n = \lim_{n \to \infty} \frac{n}{n+1} = 1.$$
   • For ( c_n ):
   $$\lim_{n \to \infty} c_n = 1.$$

4. Draw a Conclusion: Since ( b_n \leq a_n \leq c_n ) and both of them are heading toward 1, we can say that:

$$\lim_{n \to \infty} a_n = 1.$$

In short, the Squeeze Theorem shows us that by using the right sequences around a tricky sequence, we can prove that it behaves a certain way. This makes understanding sequences a lot easier!