Is there a smallest real number $a$ such that there exist a natural number $N$ so that:

$n>N\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{p}_{n+1}\le a\cdot {p}_{n}$?

I believe it can be proved that $n>7\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{p}_{n+1}\le \sqrt{2}\cdot {p}_{n}$.

$n>N\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{p}_{n+1}\le a\cdot {p}_{n}$?

I believe it can be proved that $n>7\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{p}_{n+1}\le \sqrt{2}\cdot {p}_{n}$.