test the uniform convergency of the sequence of function Test the uniform convergency of the following sequence of functions in $[0,\pi]$. $$ f_n(x)=\frac{\sin nx}{1+nx}$$ Clearly we can see that in converges pointwise to zero funcion in $[0,\pi]$. But using the definition of uniform convergence I can see that it is not uniform convergent in the given interval. Plese help me to solve it out.

If $x_n=\pi/2n$, then $f_n(x_n)=\frac{1}{(1+\pi/2)}$. Take $\epsilon = \frac{1}{2(1+\pi/2)}$.