Let Y be a random variable which takes values on the integer

Let Y be a random variable which takes values on the integers. Show that for every integer k we have

Solution

As Y only takes integer values, then

P(Y<=k) = P(Y=k) + P(Y = k-1)+ P(Y = k-2)+ P(Y = k-3)...

P(Y<=k-1) = P(Y = k-1)+ P(Y = k-2)+ P(Y = k-3)...

Subtracting them, we see that all terms cancel after P(Y =k),

P(Y = k) = P(Y<=k) - P(Y<=k-1) [DONE!]