= Monotone integral Tauberian lemma
{title2=$\int_1^\infty(\beta(x)-x)x^{-2}dx\text{ converges}\ \Longrightarrow\ \beta(x)/x\to1$}
For nondecreasing real $\beta$, convergence of the displayed improper integral forces $\beta(x)/x\to1$. For $\lambda>1$, the integral over $[X,\lambda X]$ tends to zero and is at least $(1-\lambda^{-1})\beta(X)/X-\log\lambda$. The integral over $[X/\lambda,X]$ tends to zero and is at most $(\lambda-1)\beta(X)/X-\log\lambda$. These bound the upper and lower limits by $\log\lambda/(1-\lambda^{-1})$ and $\log\lambda/(\lambda-1)$. Letting $\lambda\downarrow1$ squeezes both to one. The limit of the improper integral must be finite.
Back to article page
