[Math] Uniform Convergence of Sequence of Functions Doubt

Question

For the sequence of functions $$f_n={nx\over 1+n^3x^3},x\in[0,1]$$ If we wish to check the Uniform Convergence. I tried doing it in the following two ways

Method 1 :

$\lim f_n=0=f(x)$ , for a given $\epsilon>0$

$|f_n(x)-f(x)|=|{nx\over 1+n^3x^3}-0|={nx\over 1+n^3x^3}<{nx\over n^3x^3}={1\over n^2x^2}<\epsilon$

so we have , the result holds good for $n>{1\over x \sqrt{\epsilon}}$but, as $x\rightarrow 0, m\rightarrow\infty $. Tis means that $f_n$ is not uniformly convergent

Method 2: Using $M_n-test$

let $y={nx\over 1+n^3x^3},y'={n(1-n^3x^2)\over (1+n^3x^2)^4}$. Setting $y'=0, y={1\over n^{3/2}}$ This is the maximum value as confirmed by $y''$. As $M_n\rightarrow 0 \ as \ n\rightarrow\infty $, The function must be Uniformly convergent.

So where am i doing/understanding it wrong ?

Answer 1

HINT: The functions converge pointwise to $0$, and $$ f_n\left( \frac{1}{n} \right) = \frac{1}{2}.$$