$DTIME(f(n)) \subset of DSPACE(f(n))$
If space is greater than time, as it must surf the space (in the related Turing machine), it will be contradicted by time is less than space. Hence, we can find that all problems which are in $DTIME(f(n))$ are in $DSPACE(f(n))$. and the proof is completed.

On the other hand, the size of input is important. Hence, if we set $m = n^2$ we reach to the first relation. Hence, it is true again.