How to prove the union bound with Markov's inequality? We have events $B_1$, $B_2$, $\dots$, $B_t$. Prove $\Pr\left(\bigcup_{i=1}^t B_i\right) \le \sum_{i=1}^{t}\Pr(B_i)$.
Wikipedia proves by induction and I also understand this inequality intuitively, that is when summing all the events you're computing the overlapped events multiple times. But I'm not sure how to prove this using markov's inequality. Can someone give some insights into how to prove this? Thanks so much!

Hint: take $X = 1_{B_1} + \dots + 1_{B_t}$.  Note that $\bigcup_{i=1}^t B_i = \{X \ge 1\}$, so use Markov's inequality to estimate the probability of the latter event.
