If $L$ is an APX-hard language, doesn't the existence of a PTAS for $L$ trivially imply $\mathsf{P} = \mathsf{NP}$?Since for example metric-TSP is in APX, but it is not approximable within 220/219 of OPT [1] unless $\mathsf{P} = \mathsf{NP}$. Thus if there was a PTAS for $L$ we could reduce metric-TSP using a PTAS reduction to $L$ and thus can approximate OPT within arbitrary precision.Is my argument correct?[1] Christos H. Papadimitriou and Santosh Vempala. On the approximability Of the traveling salesman problem. Combinatorica, 26(1):101120, Feb. 2006. 
$L$ APX-hard thus PTAS for $L$ implies $\mathsf{P} = \mathsf{NP}$
complexity theory;np complete;approximation
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