Let $\mathrm{L} \in \mathrm{NTIME}(n^3)$. Since $\mathrm{NTIME}(n^3) \subseteq \mathrm{NP}$, we have that $\mathrm{L} \le_p \mathrm{3SAT}$. However, $\mathrm{3SAT} \in \mathrm{NTIME}(n)$. Hence, $\mathrm{L} \in \mathrm{NTIME}(n)$. Thus, $\mathrm{NTIME}(n^3)\subseteq \mathrm{NTIME}(n)$ which implies the non-deterministic time-hierarchy is false.
But we all know that time hierarchy is true. Where am I going wrong? The statement seems to be correct but I know it's wrong. How?