Make the $\varepsilon$–$N$ story visual. Compare $f_n$ to its pointwise limit $f$, watch $\sup_x |f_n-f|$, and test uniformity.
$$f_n(x)=x^n,\ x\in[0,1]$$pointwise, not uniform
$$f_n(x)=\\sin(nx)/n,\ x\in[0,2\\pi]$$uniform
$$f_n(x)=\\dfrac{n}{1+n^2x^2},\ x\in\\mathbb R$$fails at x=0
Current $\sup_x |f_n-f|$
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Grid resolution
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ε-test at current n
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Uniform? (theory)
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$$\textbf{Uniform vs Pointwise.}\\
(f_n) \to f \text{ pointwise if }\forall x\,\forall \varepsilon>0\,\exists N\,\forall n>N:\ |f_n(x)-f(x)|<\varepsilon.\\
(f_n) \to f \text{ uniformly if }\forall \varepsilon>0\,\exists N\,\forall n>N\,\forall x:\ |f_n(x)-f(x)|<\varepsilon. $$
$$\text{We visualize } \varepsilon \text{-bands and compute } \sup_x |f_n-f|\text{ over the chosen domain.}$$
Curves: $f_n$ (blue) vs limit $f$ (pink) with $\varepsilon$-band (shaded). Drag $\varepsilon$ to see band width change. Heatmap shows minimal $N(\varepsilon,x)$.