test the uniform convergency of the sequence of function

Question

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.

Answer 1

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

Answer 2

Hint: If we have uniform convergence, then $f_n(x_n) \longrightarrow 0$, $\forall$ $(x_n) \subset [0,\pi]$.

Choose $x_n = \frac{1}{n}$.

Answer 3

Since $f_{n}(1/n) \rightarrow \frac{sin1}{2}$ which is a positive real so sup $\{ f_{n}(x)\}$ does not tends to zero on $[0,1]$ and hence non uniform convergence.