Converges Uniformly

Let , and .
We say converges uniformly to if:

We write uniformly.

Note that if pointwise means:

So we just swapped two quantifiers.
It is now easy to see that uniform implies pointwise.

Visually, all functions need to be in this -neighbourhood of :

Theorem CTS

Suppose each is Continuous and uniformly.
Then is CTS.

Proof

Let and . By uniform convergence, find s.t.

In particular,

By continuity of at , we can find s.t.

Now let s.t. . Have

Hence is CTS.

This is called a -proof.

Theorem INT

Let the domain .
Suppose uniformly and each is Riemann Integrable.
Then is also Riemann Integrable and .

Proof

Take large enough so that .
Note that is bounded (by ).
Hence is bounded by .
Now for any dissection , and are defined.
Pick .
Hence find big enough s.t.
and find a dissection s.t. .
Furthermore, can find that and similar for

Now use triangle inequality on to find a proof.

Finally, have:

Hence .

Corollary

Suppose is a sequence of functions whose partial sums converge uniformly to .
Then:

Theorem Diff

Let be .
Assume that the sequence of partial sums of converges uniformly
and that there is some such that:

converges.
Then the sequence of partial sums converges uniformly.
Furthermore, the limit is and:

Proof sketch

Define where is the uniform limit of partial sums of .
Note that by FTA, is differentiable and it’s derivative is .
So we just need to prove that partial sums of converge uniformly to .
Let .
Find s.t. for any we have AND .
By FTA we have .
So: