# Mathematical Induction problem- how to show that $P(\bigcup_{i=1}^nA_i)\leq\sum_{i=1}^n P(A_i)$?

Hi I tried doing this problem but im not sure if I am correct can someone help

So this what I did

Now I am not so sure if this is right

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You might (instead) want to use the result $P(A\cup B)\leq P(A)+P(B)$ in the induction step. In particular, note that $$\bigcup_{i=1}^{n+1}A_i=\left(\bigcup_{i=1}^nA_i\right)\cup A_{n+1}$$ and that $$\left(\sum_{i=1}^nP(A_i)\right)+P(A_{n+1})=\sum_{i=1}^{n+1}P(A_i),$$ then use your inductive hypothesis and the previous result.
Your approach is fine, though you might want a couple extra steps in the chain of inequalities to make it clear how you got there, and you should fix your last expression to $$\sum_{i=1}^{n+1}P(A_i).$$ Your approach has the added advantage that it hints at the circumstances under which the inequality becomes equality.