Why must sin(1/x) have a zero between x = .01 and x = .001

[kevin.g...@gmail.com]

Hi,

I'm concerned about how to write a rigorous enough proof for this question. I know sin (1/x) oscillates closely to 0, but feel like that is a lacking answer.

The only other thought I have is to write something to the extent of 

if abs(1/x0 - 1/x1) > period of sine then there is at least one zero

but I don't know if that is an appropriate answer either.

Thanks,

Kevin Andrews

[William DeMeo]

Your reasoning seems sound to me. Another way to say it is 1/0.01 = 100 and 1/0.001 = 1000. Since, as you say, sin(x) has a period of 2pi, and since sin(x) crosses the x-axis twice per period, it must cross the x-axis many times over the (large) interval 100 < x < 1000.