why is hazard function tending to infinity

Question

How to prove that if h(t) hazard function is failure rate function then integration of h(t) from 0 to infinity is infinity .

Answer 1

The cumulative hazard function $\Lambda(t) = \int_{0}^t h(s) ds$ is related to the survival function by $S(t) = \exp(-\Lambda(t))$. The survival function is cadlag starting at 1 at time 0 and tending to 0 as $t \rightarrow \infty$. Therefore $\Lambda(t) \rightarrow \infty$ as $t \rightarrow \infty$.