In the book " Number Theory IV Transcendental Numbers" written by Parsin and Shafarevich (book, page 104) it is asserted that to explicit a transcendence measure of a complex number $w$, it is sufficient to prove that there is an infinite sequence of polynomials $P_m(x)\in\mathbb Z[x]$ of fixed degree such that $$0<H(P_m)\le c^m\qquad e^{-\lambda_1m}\le|P_m(w)[\le e^{-\lambda_2m}$$ where $c$, $\lambda_1$, and $\lambda_2$ are constants, such that $c > 1, \lambda_l > \lambda_2 > 0$ with $H(P_m)=\max\{|\text{coefficients of $P_m$}|\}$. I do not know why. Can anyone give the argument behind this assertion or a reference? This result is used to give a transcendence measure of $\ln(r)$ ($r$ is a rational)

Thanks in advance