Space/Time Hierarchy Theorem - proving the language of the machine is in the larger space class I am trying to understand the space and time Hierarchy theorems according to 


  Sanjeev Arora, Boaz Barak: Computational Complexity: A Modern Approach 


but the more general case.

What I don't understand is why the language decided by our custom machine is in the larger complexity class. For example, with the case of time complexity, the cost of simulating the input machine is $t_1 \log t_1$, so why is the language decided by it in $DTIME(t_2)$ but not in $DTIME(t_1 \log t_1)$?

thanks
A rich man walks into a bar and says, "I can buy this beer because I am a millionaire." Eloo says, "The beer costs \$5. I don't understand why this millionaire can buy it, but this guy with five dollars can't."

Remember that $\mathrm{DTIME}[f(n)]$ is the class of languages that can be decided by a deterministic Turing machine that uses at most $f(n)$ steps on any input of length&nbsp;$n$. So, since $t_1\log t_1\leq t_2$, anything in $\mathrm{DTIME}[t_1\log t_1]$ is in $\mathrm{DTIME}[t_2]$.