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 $$EXPSPACE$$.

Question

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