599

证明：由于$\lim \limits_{x \to \begin{array}{*{20}{c}}{{\rm{ + }}\infty }\end{array}} f\left( x \right){\rm{ = }}0$，则对任给的$\varepsilon  > 0$，存在$M>0$，使得当$x > M$时，有$\left| {f\left( x \right)} \right| < \varepsilon$

特别地，取$\varepsilon  = 1$，则当$x > M$时，有$\left| {f\left( x \right)} \right| < 1$，于是有${f^2}\left( x \right) \le \left| {f\left( x \right)} \right|$

又知$\int_a^{ + \infty } {f\left( x \right)dx}$绝对收敛，从而由比较判别法知$\int_a^{ + \infty } {{f^2}\left( x \right)dx}$收敛