Sequences that are subsequences of each other I've been thinking about this problem for awhile now, but I still can't figure out part (b)... Any help would be greatly appreciated! 
(a) Find two distinct sequences $a_n$ and $b_n$ such that $a_n$ is a subsequence of $b_n$ and $b_n$ is a subsequence of $a_n$.
(b) What if we also enforce the condition that $\{ a_n | n \in \mathbb{N} \}$ must be infinite? 
I'm understanding "infinite" as unbounded right now. I don't know if that's actually the case.

For (a), if $a_n$ and $b_n$ oscillate in some different way between two values, so that neither is eventually constant, you are there.  Can you see why?
The condition in (b) demands that $a_n$ must take an infinite number of distinct values.  You are correct that since the values are in $\Bbb N$ that means $a_n$ must be unbounded, but if the values were in $\Bbb R$ it would not.  For this problem it is not important.  
You can get halfway there by making $b_{n+1}=a_n$ and choosing $b_1$ to be whatever you want.  Clearly $a_n$ is a subsequence of $b_n$.  Now if $a_n$ consists of a series of ramps, each higher than the last, then returns to $1$, so $1,2,1,2,3,4,5,1,2,3,4,5,6,\dots$ you are home.
