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.
Hint: Assume it has a bound $A$, then show that it will eventually exceed that bound.
How much do you add at each step? If there is some positive integer $n$ such that you always add at least $1/n$, then obviously the sequence grows without bound.
If there is no such number, then for any $n$ you will eventually add less than $1/n$. The only way to add less than $1/n$ is if your sequence grows to more than $n$. So your sequence must grow without bound in this case too.
