I've been working on proving that there is always a prime between $n$ and $2n$, and also that there is always a prime between $n^2$ and $(n+1)^2$ (Legendre's conjecture).
I believe I've proven those two statements using the following fact:
The most consecutive integers divisible by a number less than or equal to $n$ is $p-2$, where $p$ is the first prime larger than $n$.
For example, if $n=8$ then $p=11$ and I can find at most $9$ consecutive integers divisible by numbers less than $8$. ($200$ through $208$ is one example of consecutive integers divisible by $2,3,5,$ or $7$)
My question is if this fact about consecutive integers divisibility is well known in number theory, or if it is something I will have to prove separately before the other two proofs are valid?
Update: This was proven false in the answer below by Qiaochu Yuan.