Answer to Question #346719 in Real Analysis for Nikhil Singh

Question

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

Answer 1

The given sequence is pointwise convergent on [-2,2], since

$\lim\limits_{n\to\infty}f_n(x)=\lim\limits_{n\to\infty}\frac{nx}{1+nx^2}=\lim\limits_{n\to\infty}\frac{x}{x^2+\frac{1}{n}}=\begin{cases} \frac{1}{x}, & \text{if }x\ne 0\\ 0, & \text{if }x=0 \end{cases}:= f(x)$

Weierstrass Theorem claims that the limit of uniformly convergent sequence of continuous functions must be a continuous function. All the functions $f_n(x)$ are continuous on $[-2,2]$, but $f(x)$ is not, therefore, by Weierstrass theorem, the convergence of $f_n(x)$ to $f(x)$ is not uniform on [-2,2].