A note on gaps

Let $p_{k}$ denote the $k$-th prime and $d(p_{k}) = p_{k} - p_{k - 1}$, the difference between consecutive primes. We denote by $N_{\epsilon}(x)$ the number of primes $\leq x$ which satisfy the inequality $d(p_{k}) \leq (\log p_{k})^{2 + \epsilon}$, where $\epsilon > 0$ is arbitrary and fixed, and by $\pi(x)$ the number of primes less than or equal to $x$. In this paper, we first prove a theorem that $\lim_{x \to \infty} N_{\epsilon}(x)/\pi(x) = 1$. A corollary to the proof of the theorem concerning gaps between consecutive squarefree numbers is stated.