Uniform convergence of a family of functions on $(0,1)$

Question

Let the family of functions be

$$f_n(x) = \dfrac{x}{1+nx}.$$

Is the sequence $f_n$ uniformly convergent in the interval $(0,1)$?

Answer 1

$\frac{x}{1 + nx} = \frac{1}{\frac{1}{x} + n} \leq \frac{1}{n}$ which doesn't depend on $x$ hence your sequence converges uniformly to $0$

Answer 2

At first we have that $f_n\to 0$. Now $\|f_n\|_{\infty}=sup|f_n|=supf_n=1/(1+n)\to 0$ and thus is uniformly convergent in $(0,1)$.

$f'_n(x)=\frac {1}{(1+nx)^2}$ and thus $f_n$ are stricklty increasing and thus $supf_n(x)=f_n(1)$