Lagmental Vicfred

Uniform Limits of Continuous Functions Are Continuous by Vicfred

tags: mathematics, uniform convergence, compactness, approximation

A single uniform error bound combines with continuity of one approximant to control the limit. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Statement

A sequence \(f_n:X\to\mathbf R\) converges uniformly to \(f\) when \(\sup_{x\in X}|f_n(x)-f(x)|\to0\). The supremum norm captures one error bound valid everywhere.

$$ f_n\in C(X),\qquad\|f_n-f\|_\infty\to0 $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ f\in C(X) $$

Worked algebra

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ |f(x)-f(x_0)|\le\underbrace{|f(x)-f_N(x)|}_{<\varepsilon/3}+\underbrace{|f_N(x)-f_N(x_0)|}_{<\varepsilon/3}+\underbrace{|f_N(x_0)-f(x_0)|}_{<\varepsilon/3} $$

Conceptual compression

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad f_n\in C(X),\qquad\|f_n-f\|_\infty\to0,\\[5pt] \mathsf{C}\;&:\quad f\in C(X). \end{aligned} $$

Caveat

Pointwise convergence allows the bad point to move with \(n\). Continuity, integration, and differentiation survive limits under different uniform hypotheses.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] f\in C(X) \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Sat 16 August 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.