Answer to Question #348644 in Real Analysis for Sarita bartwal

Question

Prove that the sequence (fn(x)), where fn(x)= nx/(1+ nx^2) is not uniformly convergent in [-2,2]

Answer 1

PROOF

Let $x=0,$ then $f_{n}(0)=0$ for all $n\in\N$ and $\lim_{n\rightarrow\infty}f_{n}(0)=0$ . If $x\neq0$ ,then $\lim_{ n \rightarrow\infty} f_{n}(x)=\lim_{n\rightarrow\infty} f_{n}(x)=\lim_{n\rightarrow\infty}\frac{nx} {1+nx^{2}}=\lim_{n\rightarrow \infty}\frac{nx} {n(\frac{1}{n}+ x^{2})}=\lim_{n\rightarrow\infty}\frac{ x} { (\frac{1}{n}+ x^{2})}=\\=\frac{x}{x^{2}}=\frac{1}{x} .$

Thus $f_{n}(x)$ converges pointwise to

$f(x)= \begin{cases} 0& \text{ if } x=0 \\ \frac{1}{x}& \text{ if } x\in [-2,0)\bigcup (0,2] \end{cases}$ .

Note that the functions $f_{n}(x)$ are continuous on the segment $[-2,2]$ , the function $f$ is discontinuous. Therefore, by Theorem 1 (see bellow) $f_{n}(x)$ does not converge uniformly on $[-2,2]$ .

THEOREM 1. Let $(f_{n}(x))$ be a sequence of function on $A$ converging uniformly to $f$ on $A$ .If each of $f_{n}$ is continuous on $A$ , then $f$ is continuous on $A$