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# Uniform Convergence Wall Art Poster

## Uniform Convergence Wall Art Poster

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An early conceptual hurdle for students of analysis is convergence of sequences of numbers. A subsequent hurdle arises when we consider sequences of functions $$(f_{n})$$: There is more than one way a sequence of functions can converge!
Asked to define "convergence of a sequence of functions," many students would write down "pointwise" convergence of the numerical sequence $$\bigl(f_{n}(x)\bigr)$$ for each $$x$$ in the common domain of the $$f_{n}$$. Unfortunately, a sequence of continuous functions may converge pointwise to a discontinuous limit. For example, if $$f_{n}(x) = x^{n}$$, then each $$f_{n}$$ is continuous on $$[0, 1]$$. On the other hand, $$f_{n}(x) \to 0$$ if $$0 \leq x < 1$$ while $$f_{n}(1) = 1 \to 1$$; the limit function is discontinuous.
"Uniform convergence" on a domain is a stronger condition that more closely conforms to students' intuition. Unfortunately, distinguishing uniform convergence from pointwise convergence is technically subtle. This poster image depicts the difference visually in a prototypical example, essentially the functions $$f_{n}(x) = x^{n}$$ for $$0 \leq x \leq 1$$.