I mostly see stopping times used to stop a random process at time $\tau$. However, in some proofs, the following identity is used, which I cannot deduce myself:
Let $\tau$ be a stopping time. Why $$ E(\tau) = \sum _{i=1}^\infty P(\tau \geq i). $$ $\tau$ is a discrete stopping time, so why not $$ E(\tau) = \sum _{i=1}^\infty iP(\tau = i). $$