The bounded monotone theorem says a sequence converges. Sometimes mathematicians only care if the sequence converges or diverges. They do not care what it converges to. Sometimes it can be really hard to find what it converges to. Thus, the theorem tells us something converges without the need to find the number.

A stupid example would be . This sequence is monotone (why?) and bounded (why?) thus it is convergent. In this case it is easy to see convergence.

Here is an application of squeeze theorem. Say we want to find the limit of . We note that . The outer sequences both converges to 0 thus the inner sequence converges to 0.