TITLE: A question which is to be proved assuming Bertrand Postulate
QUESTION [0 upvotes]: I am trying  exercise problem of David Burton Elementary Number Theory and I am unable to think about this problem. 

Problem is->Prove that for every n$\geq$ 2 there exists a prime p with p$\leq$n< 2p . 

Can someone please help how to think about this problem. 
I shall be really thankful.

REPLY [1 votes]: According to Bertrand’s postulate, there is a prime $p$ with $k\lt p\lt2k$ for all integers $k\gt1$. Apply this for $k=\left\lceil\frac n2\right\rceil$ to obtain a prime $p$ with $\left\lceil\frac n2\right\rceil\lt p\lt2\left\lceil\frac n2\right\rceil$ and thus $\frac n2\lt p\le n$. Now multiply the left inequality by $2$ and move it to the right.
(This proof doesn’t apply for $n=2$, since in that case $\left\lceil\frac n2\right\rceil=1$, so we have to check separately that there is a prime $p=2$ such that $p\le2\lt2p$.)
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