Why is my proof wrong? (analysis)

[Artusartos]

I was trying to prove this, but something went wrong and I'm not sure what...

We know that the greatest value of sin(1/x) can be 1 and the lowest value can be -1...

We know that the upper darboux sum is...

$$U(f,P) = sup(f[t_{k-1} , t_t]) (t_k - t_{k-1}) = (1) (1-(-1)) = 2$$


We also know that the lower darboux sum is...

$$L(f,P) = inf(f[t_{k-1} , t_t]) (t_k - t_{k-1}) = (-1) (1-(-1)) = -12$$

So we know that...

$$U(f,P) - L(f,P) = 2 - (-2) = 4 > \epsilon$$

Can anybody tell me where I went wrong?

Thanks in advance

 

[micromass]

What are you trying to prove in the first place??

And don't there need to be summations in the $U(f,P)$ and $L(f,P)$?? How did you choose P anyway?

I also don't see where -12 comes from. Did you mean -2?

What is $\epsilon$ supposed to be?