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

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

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

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