I am self studying some number theory from notes and I am unable to deduce following result.
Chebyshev's Theorem : There exists constants c<1<C such that $cx /\log x \leq \pi(x) \leq Cx/\log x$.
It is written that Chebyshev's estimate is equivalent to $c' x \leq \psi(x) \leq C' x$ .
I thought of using the result that lim x tending to $\infty$ $\frac{\psi(x) }{x} = $ lim x tending to $\infty$ $\frac{\pi(x) log x}{x}$.
But this proves the result for only when x tends to $\infty$ ,
How can I prove equivalence for any x? Can you please tell.