Why is the language $NP-hard$?
You reduce $\text{gap-IS}[1/8+\epsilon,1/4-\epsilon]$ (for any $\epsilon &lt; 1/20$) to $A$. The reduction is the identity function. Let's see why it works:


If a graph $G$ is a YES instance of $\text{gap-IS}[1/8+\epsilon,1/4-\epsilon]$ then it has an independent set of size at least $(1/4-\epsilon)n \geq n/5$.
If a graph $G$ is a NO instance of $\text{gap-IS}[1/8+\epsilon,1/4-\epsilon]$ then it has no independent set of size $(1/8+\epsilon)n &lt; n/5$.


Hence a graph $G$ is a YES instance of $\text{gap-IS}[1/8+\epsilon,1/4-\epsilon]$ if and only if it is a YES instance of $A$.