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] Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM 45, 3 (May 1998), 501-555.

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):101–120, Feb. 2006.

If $L$ is an APX-hard language, doesn't the existence of a PTAS for $L$ trivially imply $P =NP$?

Since for example metric-TSP is in APX, but it is not approximable within 220/219 of OPT unless P = NP, thus a 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?

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] Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM 45, 3 (May 1998), 501-555.