Question on time hierarchy
A language is in $\mathsf{DTIME}(f(n))$ if there is a Turing machine accepting the language and running in time at most $f(n)$ (or, in some definitions, $O(f(n))$). Therefore $f(n) \leq g(n)$ implies that $\mathsf{DTIME}(f(n)) \subseteq \mathsf{DTIME}(g(n))$. The same holds for relativized classes.