Boundedness of $a+\frac 1a$ when iterated Here's something I was wondering... 

Is $$a + \frac 1a$$ for any positive real number $a$ bounded when iterated? 

For example,, if we start at $a=1$, continuing gives us $a= 1+ \frac 11=2$, then $a=2+\frac 12=2.5$ and so on. A quick program shows that it seems to grow without bound, but how would one prove this mathematically? If it is possible that is... Any hints would be appreciated.

The function
$$
f(x)=x+\frac1x
$$
is strictly increasing for $x\ge1$ (i.e. if $x>y$, then $f(x)>f(y)$). Also, we have that
$$
f(x)=x+\frac1x>x
$$
for $x\ge1$. Hence, $f(f(x))>f(x)$. However, the function might be bounded, i.e. $f(x)\le M$ for all $x\ge1$. But we have that $f(M)>M$, which is a contradiction. So it grows to infinity.
