Take a subsequence converging to 2008, and one converging to 2009. Argue that you may assume that they have no terms in common, and in fact, the terms in the first subsequence are "very close" to 2008, and those in the second are "very close" to 2009 (if $b_n\to L$ then for $n$ large, $|b_n-L|<1/8$, and we can restrict ourselves to large values of $n$ in both subsequences).
Argue that you may assume the indices of the sequences alternate. This means that if $a_{n_1}, a_{n_2},\dots $ is the sequence converging to 2008 and $a_{m_1},a_{m_2},\dots$ is the sequence converging to 2009, then we may assume that $n_1 (for this, all you need is to "thin" out your subsequences).
Ok. Now argue that for all $i$, between $n_i$ and $m_i$ there is some $k_i$ such that both $a_{k_i}-a_{n_i}$ and $a_{m_i}-a_{k_i}$ are $\ge1/4$. It is here that you use that $|a_{n+1}-a_n|\le1/2$ for all $n$, noting that 2008 and 2009 are 1 unit apart, and $|a_{n_i}-2008|\le 1/8$ and $|a_{m_i}-2009|\le 1/8$.
Now, the sequence $(a_{k_i}\mid i=1,2,\dots)$ is completely contained in the interval ${}[2008.25,2008.75]$, so it must have a convergent subsequence, and this subsequence must converge somewhere other than 2008 and 2009.
On the other hand, note that you cannot conclude that there are more than 3 "partial limits", by considering the sequence $2008, 2008.5, 2009, 2008.5, 2008, 2008.5,\dots$