
Equicontinuous
Prove or disprove
$\displaystyle F_n(x) = \sin nx $ is equicontinuous
I know the definition of equicontinuous at $\displaystyle x_0$ it says for all $\displaystyle \epsilon >0 $ there exist $\displaystyle \delta>0 $ such that if $\displaystyle d ( f(x_0),f(x) ) < \epsilon $ then
$\displaystyle d(x_0 , x) < \delta $
trying if it is equicontinuous at $\displaystyle x_0 = 0 $
Given $\displaystyle \epsilon > 0 $
$\displaystyle  f(x)  < \epsilon \Rightarrow \sin n x  < \epsilon $
delta depends on epsilon and x just how i can continue
any hints or any directions

Re: Equicontinuous
Do we have to decide whether the family is equicontinuous at each point, or to determine the point on which the family is equicontinuous? For the first question, look at $\displaystyle F_n\left(\frac{\pi}{2n}\right)$.