# Interactive Real Analysis - part of MathCS.org

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## Definition 8.2.1: Uniform Convergence

A sequence of functions { fn(x) } with domain D converges uniformly to a function f(x) if given any > 0 there is a positive integer N such that
| fn(x) - f(x) | < for all x D whenever n N

Please note that the above inequality must hold for all x in the domain, and that the integer N depends only on .

We should compare uniform with pointwise convergence:

• For pointwise convergence we could first fix a value for x and then choose N. Consequently, N depends on both and x.
• For uniform convergence fn(x) must be uniformly close to f(x) for all x in the domain. Thus N only depends on but not on x.

Let's illustrate the difference between pointwise and uniform convergence graphically:

Pointwise Convergence Uniform Convergence

For pointwise convergence we first fix a value x0. Then we choose an arbitrary neighborhood around f(x0), which corresponds to a vertical interval centered at f(x0).

Finally we pick N so that fn(x0) intersects the vertical line x = x0 inside the interval (f(x0) - , f(x0) + )

For uniform convergence we draw an -neighborhood around the entire limit function f, which results in an "-strip" with f(x) in the middle.

Now we pick N so that fn(x) is completely inside that strip for all x in the domain.

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