Squeeze Theorem

analysis

Definition

Squeeze Theorem

Let and be convergent sequences with identical limits . Let be a sequence such that:

for almost all . Then is also convergent, and its limit is equal to :

Function Version

The theorem applies analogously to limits of functions:
If for all in a neighbourhood of (except possibly at ), and , then .

Proof

Squeeze Theorem

Let and be convergent sequences with identical limits . Let be a sequence with .

For any , by the definition of convergence:

1. There exists such that for all ,, which implies .
2. There exists such that for all ,, which implies .

Let . Then for all :

Thus, , which means . This proves that .