Are there any prime gap results like this

[uart]

Are there any "prime gap" results like this ...

I was just reading about "prime gaps" and noticed that most of the results are asymptotic, as in "true if n is sufficiently large".

I was just wondering if there are any bounding results for prime gaps that are true for all n, p_n.

For example, take a conjecture like: $$p_{n} < p_{n+1} < 2 p_{n}$$

Is something like that provable for all n. (not necessarily with the constant of "2", I just chose that as an example of what I meant).

 

[shmoe]

With a "2", see Bertrand's postulate.

Some explicit ones (specifying "n large enough") can be found on http://primes.utm.edu/notes/gaps.html

You might also want to look at http://math.univ-lille1.fr/~ramare/Maths/gap.pdf

 

[uart]

With a "2", see Bertrand's postulate.

Thanks, that was just what I was looking for but I didn't have a name to search on. I thought that someone would have postulated it before me. :)