Let $\{f_n\}^{+∞}_{n=1}$ be a sequence of functions where $f_{n}(x)=\frac{nx}{1+n+x}$ where x belongs to [0,1] and n belongs to N (all natural numbers).
Determine whether the sequences of functions $\{f_n\}$ is uniformly convergent or not.
I've found the point-wise limit is $$ f(x)=\left \{\begin{array}{l} 1, x=1 \\ x,0<x<1\\ 0, x=0 \end{array} \right. $$
But I don't know I should choose which function to compute $|f_n(x)-f(x)|$ and find the supremum of $|f_n(x)-f(x)|$
The convergence is uniform. Note that $|f_n(x)-f(x)|=\frac {x(1+x)} {1+n+x} \leq \frac 2 {1+n}$ for $0<x<1$. Also, $|f_n(x)-f(x)|\to 0$ for $x=0$ and $x=1$. These facts imply that $f_n \to f$ uniformly.