Does sinx < x hold true for all positive values of x?

[Amer]

My work it is clear that
x = sin x, at x = 0
if we look at the slope of each curve

1 , cos x
cos x <= 1
which means sin(x) will go down and x will remain at it is direction, and the two curves will never met unless the slope of sin(x) be more than 1
How about it ?
Thanks in advance

 

[tkhunny]

It is never good to start with "It is clear that...". Really, just never do that. Simply state what you wish to state. No need to insult your audience.

Your argument is fuzzy, but you are on the right track.

1) I am tempted to limit the scope of the proof by pointing out one thing before I start the actual proof. $$-1 \le \sin(x) \le 1 \therefore$$, For $$x>1$$, we have $$x > \sin(x)$$.

Now we can focus on $$0 < x \le 1$$.

2) I might be tempted to to ponder $$f(x) = x - \sin(x)$$. There must be an Intermediate Value Theorem, or something, in there.