  In contemporary research, data scientists often test an infinite sequence of
hypotheses $H_1,H_2,\ldots $ one by one, and are required to make real-time
decisions without knowing the future hypotheses or data. In this paper, we
consider such an online multiple testing problem with the goal of providing
simultaneous lower bounds for the number of true discoveries in data-adaptively
chosen rejection sets. Using the (online) closure principle, we show that for
this task it is necessary to use an anytime-valid test for each intersection
hypothesis. Motivated by this result, we construct a new online closed testing
procedure and a corresponding short-cut with a true discovery guarantee based
on multiplying sequential e-values. This general but simple procedure gives
uniform improvements over existing methods but also allows to construct
entirely new and powerful procedures. In addition, we introduce new ideas for
hedging and boosting of sequential e-values that provably increase power.
Finally, we also propose the first online true discovery procedure for
arbitrarily dependent e-values.
