Convergence of sin(x/n) in A[0,pi]

[desaila]

Homework Statement

Show that the sequence real-valued functions fn(x) = sin(x/n) converges uniformly to f(x) = 0 on A=[0,pi]

Homework Equations

I don't believe there are any

The Attempt at a Solution

Well, my first thought was that if x/n = pi then sin(x/n) is obviously going to be zero. I really can't get past that, I mean it seems like stating that and that if x/n is any multiple of pi then sin(x/n) will be zero. I guess I can't see past the obvious and into how this problem might actually be challenging.

 

[Avodyne]

You are supposed to show that fn(x) goes to zero as n goes to infinity for *any* value of x between 0 and pi.

 

[desaila]

Oh. Well then. You can't use euler's because the function is real valued, correct? Or, can you expand it and just leave off the imaginary part?

I can't think of another way to show that this function is going to zero.

 

[Office_Shredder]

Well, as n goes to infinity, x/n goes to zero for all x on [0,pi] And sin(0)=0

 

[ZioX]

If the sup norm ( sup |fn(x)-f(x)|, over x in X) goes to zero then f_n converges uniformly to f on X. This is biconditional.

I suspect you can also show this from the definition of uniform convergence.