The characteristic-function interval inequality for primes

Let $\pi(x)$ be the prime-counting function. For a positive integer $c$, let $I_k=[(k-1)c,kc)$ and define $\chi_{I_k}=1$ if $I_k$ contains one or more primes, and $\chi_{I_k}=0$ otherwise. Characteristic-function interval inequality. For any $m_2>m_1>0$, with $m_2$ and $m_1$ sufficiently large, and any $c>2$, one has

$$\frac{1}{c}\bigl(\pi(m_2c)-\pi(m_1c)\bigr)\leqslant\sum_{k=m_1}^{m_2}\chi_{I_k}<\pi(m_2c)-\pi(m_1c).$$

The inequality follows directly from counting prime-containing intervals, and the supplied text presents the claim without a conjectural status or resolution; its formulation is closely related to an elementary proposition earlier in the paper.