Question on deduction that a certain problem requires exponential space
Here's one simple resolution. The statement referred to as "Fact 2" is a slightly weaker form of the standard Space Hierarchy Theorem. The standard version of the Space Hierarchy Theorem has the quantifiers rearranged: 


  For every space-constructible $S_2(n)$, there exists $L \in \mathbf{DSPACE}(S_2(n))$ such that for every space constructible
  $S_1(n) \leq o(S_2(n))$, $L \not \in \mathbf{DSPACE}(S_1(n))$.