Monotone integral Tauberian lemma (${\int }_1^{\infty }(\beta (x)-x)x^{-2}dx\text{ converges}⟹\beta (x)/x\to 1$)

For nondecreasing real $\beta$, convergence of the displayed improper integral forces $\beta (x)/x\to 1$. 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 $\lambda \downarrow 1$ squeezes both to one. The limit of the improper integral must be finite.