We defined the real numbers to be a complete uniform space, meaning that limits of sequences are convergent if and only if they are Cauchy. Let’s write these two out in full:
- A sequence is convergent if there is some so that for every there is an such that implies .
- A sequence is Cauchy if for every there is an such that and implies .
See how similar the two definitions are. Convergent means that the points of the sequence are getting closer and closer to some fixed . Cauchy means that the points of the sequence are getting closer to each other.
Now there’s no reason we can’t try the same thing when we’re taking the limit of a function at . In fact, the definition of convergence of such a limit is already pretty close to the above definition. How can we translate the Cauchy condition? Simple. We just require that for every there exist some so that for any two points we have .
So let’s consider a function defined in the ray . If the limit exists, with value , then for every there is an so that implies . Then taking as well, we see that
and so the Cauchy condition holds.
Now let’s assume that the Cauchy condition holds. Define the sequence . This is now a Cauchy sequence, and so it converges to a limit , which I assert is also the limit of . Given an , choose an so that
- for any two points and above
Just take a for each condition, and go with the larger one. In fact, we may as well round up so that for some natural number . Then for any we have
and so the limit at infinity exists.
In the particular case of an improper integral, we have . Then . Our condition then reads:
For every there is a so that implies .