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 f_n(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.