Let $(g_{k})_{k≥1}$ be the sequence of primes gap. In 1938, Robert Rankin proved the existence of a constant $c > 0$ such that the inequality:
$$g_{k}>c×(log k×loglog k×logloglog k)/((logloglog k)²).....(*)$$ holds for infinitely many values $k$, improving the results of Westzynthius and Paul Erdős (https://en.wikipedia.org/wiki/Prime_gap#Lower_bounds). He later showed that one can take any constant $0<c < e^{γ}$, where $γ$ is the Euler–Mascheroni constant.
My question is: How one can show that inequality $(*)$ holds true for infinitely many primes.