# Technique for invalidating a proof

My homework assignment is to show that the Space Hierarchy Theorem's proof (as stated in Sipser's Introduction to the Theory of Computation) is invalidated by removing some part of the algorithm -- let's call it a "safeguard".

Sisper's proof goes over the algorithm $D$, stage by stage, showing what's the rationale behind every stage. Finally, he proves that $D$ (which decides language $A$) is not in space $g(n)$ (where $g(n)=o(f(n))$). He does this by assuming for contradiction there exists $M$ in space $g(n)$ that decides language $A$, then showing such $M$ would end up deciding a different language than $D$.

I'm stuck at picking a proof technique. Sipser's proof just covers all bases, but actually constructing such a TM $M$ is a different kind of task he doesn't tackle. When I modify the proof to remove one of his "safeguards", I'm having a hard time showing there actually exists that thing he safeguarded against. I guess it's one of those cases where proving is easier than finding a counterexample?...

P.S. Looking for a hint, not to cheat :)

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