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.

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.
