Define $f_n, f:\mathbb{R} \to \mathbb{R}$ as $$f_n(x) = \begin{cases} -1, &x \in \left(-\infty, -\frac{1}{n}\right)\\ nx, &x \in \left[-\frac{1}{n}, \frac{1}{n}\right]\\ 1, &x \in \left(\frac{1}{n}, \infty\right) \end{cases}$$ and $$f(x) = \begin{cases} -1, &x \in \left(-\infty, 0\right)\\ 1, &x \in \left[0, \infty\right) \end{cases} $$ Does $f_n$ converge to $f$? If not, how do I modify $f_n$ so that it converges to $f$?
2026-04-04 02:28:55.1775269735
Convergence to $\operatorname{sgn}(x)$
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