Hardness of approximation statement clarification?
Since $1 + \log n = (1 + o(1)) \log n$, for every $\beta &lt; \alpha$ you can find $N$ such that for $n \geq N$,
$$ (1-\alpha)(1 + \log n) \leq (1-\beta) \log n. $$
This suffices to reach a contradiction.