Answer
False
Work Step by Step
Since, it is true that $\int _{a}^{\infty}f(x)dx\leq \int_{a}^{\infty}g(x)$ if $f(x) \leq g(x) $ on $[0,\infty ) $.
However, something ''less than'' divergent does not mean divergent , it can also be convergent.
Hence, the given statement is false.
