# If a problem is inapproximable for $(2-\epsilon)$, can we conclude there exists no PTAS for it?

If we prove that:

The existance of a $$(2-\epsilon)$$-approximation algorithm for Problem P1 implies $$P = NP$$,

can we conclude:

There exists no PTAS for Problem P1, and so P1 is APX-hard?

You can conclude There exists no PTAS for Problem P1 if $$P \neq NP$$, but you can NOT conclude P1 is APX-hard. Precisely:
1. If someone proofs $$P = NP$$ you get a trivial PTAS;
2. Assumung $$P \neq NP$$, it still could be APX-intermediate (see https://en.wikipedia.org/wiki/APX#APX-intermediate).