I studied proof of Bertrand Postulate from M Ram Murthy Problems in analytic number theory and completely understood it .
In M Ram Murthy Book , Statement of Bertrand Postulate is (1) – For n sufficiently large , there exists a prime between n and 2n.
But while I was looking at Book Introduction to sieve methods and applications by Ram Murthy the statement of Bertrand Postulate is (2) – For every n $\geq$ 1 , there always exists a prime between n and 2n .
Can someone please tell how To deduce 2nd statement from statement 1 ie to prove that for each n $ \geq $ 1 , there exists a prime between n and 2n .
Best Answer
Statement 1 probably tells you something like "if $n \geq 750$, then there is a prime between $n$ and $2n$". That's the bound that I was taught, where my proof ultimately hit the inequality $\frac{n \log 4}{3} < (2 + \sqrt{2n}) \log(2n)$ that was required to fail.
Now you can obtain statement 2 by checking that there is a prime between $2$ and $4$, between $3$ and $6$, …, between $749$ and $1498$.