gen-explain-uniform-convergence-as-you-would-to-a-capable-studen-69c0

Explain uniform convergence as you would to a capable student, in 3-4 sentences.

# Uniform Convergence Uniform convergence means that a sequence of functions converges to a limit function at a rate that doesn't depend on where you are in the domain. While pointwise convergence allows different points to converge at different speeds, uniform convergence requires that beyond some point in the sequence, *all* functions stay within the same small distance ε from the limit function *simultaneously*. You can think of it as the graphs of the functions forming a "tube" of width 2ε around the limit function, where the entire graph eventually fits inside this tube—not just each individual point wiggling its way toward the limit independently.