Given a prime number $p_0$, by Bertrand's postulate we know that \begin{gather} p_1\ge\frac{p_0}{2}\\ p_2\ge\frac{p_1}{2}\ge\frac{p_0}{2^2}\\ \vdots\\ p_k\ge\frac{p_0}{2^k} \end{gather} where $p_1,p_2,\dots$ are prime numbers immediately preceding $p_0$.

Question.Is there a similar nice upper bound for the $p_i$?

A trivial upper bound would be $p_i\le p_0-2i$, but that is trivial and bad in the sense that for large $i$ this doesn't give a good approximation (since one or two primes may differ by $2$, but $10$ consecutive primes can't all differ by $2$, there must be something better). I am not claiming that the above result due to Bertrand gives a good approximation, but that at least gives exponential bounds, unlike in this case.