Pointwise vs. Uniform Convergence | What it means for a sequence of functions to converge
We know what it means for a sequence of numbers to converge, but what about a sequence of functions? In this video we build up the two most important answers to that question from scratch, visually.
We start with a concrete example, fₙ(x) = xⁿ, and ask: in what sense do
these functions converge to a limit? The first answer is pointwise convergence:
fix any point in the domain, evaluate every function there, and check whether
the resulting sequence of numbers converges. We check this one point at a time.
But pointwise convergence turns out not to be strong enough for many purposes. That leads us to uniform convergence, where we ask something stricter: does the whole graph of fₙ eventually fit inside an arbitrarily thin band around the limit function, everywhere at once?
We compare both notions side by side and close with the reason mathematicians care: uniform convergence preserves continuity and allows us to interchange
limits and integrals.
Next video: the consequences of uniform convergence in detail.
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