TITLE: Show that $f(x) = x$ if $f(f(f(x))) = x$.
QUESTION [12 upvotes]: If $f: \mathbb{R} \to \mathbb{R}$ and $f$ is strictly increasing, show that $f(x) = x$ if $f(f(f(x))) = x$. 
So this compulsorily ESTABLISHES that $f(x) = x$ only, and no other solution. So, merely substituting $f(x) = x$ and hence showing the given equality holds will not earn any credit.
I was proceeding via inverses, but then I got confused with the notation. And for that, the problem got tricky.

REPLY [6 votes]: If f is strictly increasing, then if x>y, f(x)>f(y).  Suppose f(f(f(x)))=f(x).  
Case 1: Suppose $f(x) < x$.  So, $x>f(x)$.  Thus, $f(x)>f(f(x))$ since $f$ is strictly increasing.  It then would follow that $f(f(f(x)))>f(f(x))$, since $f(f(f(x)))=f(x)$.  But, since f is strictly increasing we can also infer that $f(f(x))>f(f(f(x)))$.  Thus we have a contradiction and it is not the case that $f(x) < x$.
Case 2: Suppose $f(x)>x$.  The reasoning of this case comes as similar to that of case 1.  So, it does not hold that $f(x)>x$.
By trichotomy it follows that $f(x)=x$