Space complexity problem, relation between $DSPACE(log^kn)$ and $DSPACE(log^{k+1}n)$ I need help with the following:
Let $k\in \mathbb{N}$, define:
$L^k=DSPACE(O(log^k(n)))$
$NL^k=NSPACE(O(log^k(n)))$
and:
$PolyL=\bigcup_{k=1}^{\infty}L^k$
$PolyNL=\bigcup_{k=1}^{\infty}NL^k$  

I need to prove, disprove or to determine if it is an open question:
1. For every $k\in \mathbb{N}$ it holds that $NL^k\subseteq L^{2k}$
2. There exist $k\in \mathbb{N}$ s.t. $L^k=L^{k+1}$  

The first one is easy, since it is almost immediately the result of Savitch's theorem.
The second one I'm not sure...  I'd love to get some lead there.
Use the space hierarchy theorem, which is a finite form of the undecidability of the halting problem.