Is my reasoning wrong that $PSPACE$ should not equal $EXPTIME$?

It’s impossible for a problem to require exponential space without being exponential-time.

  1. Consider that if an $ EXPSPACE~~complete$ problem can be solved in $ 2^n$ time. It will now fall into the class $ EXPTIME$ .
  2. Then $ EXPSPACE~~complete$ problems are in $ EXP$ if they can be solved in $ 2^n$ time. This means they can reduce into $ EXP~~complete$ problems and vice versa.

To me, this should be easy to write a proof that $ EXPTIME$ = $ EXPSPACE$ .

My intuition tells me that if $ Exptime$ = $ Expspace$ ; then $ PSPACE$ != $ EXPTIME$ ,

Because $ PSPACE$ already is not equal to $ EXPSPACE$ .


As an amateur, what would make this reasoning be wrong or right?