show that the outcome of a probability function applied to an infinite sample space is equal to 1 As I am reading through the first chapter of my mathematic statistics book, they define the probability function as having three mandatory properties. One of these properties is that if $\mathcal{C}$ is the sample space of all possible outcomes of an expirement, then $P(\mathcal{C})=1$ One of the exercises is asking me to show that this holds true for an infinite sample space, for example: A quarter is flipped until a heads appears, and therefore the sample space is $\mathcal{C}=\\{H,TH,TTH,\dots\\}$. How could I show this to be true? A basic proof would be required, as I do not have an extensive mathematic background and this is a statistics course.

Let $P(H) = p = 1 - q = 1- P(T)$.

Let $X$ be a random variable indicating the flip when $H$ occurs.

$$\sum_{i=1}^{\infty}P(X=i) = \sum_{i=1}^{\infty}q^{i-1}p = \frac{p}{1-q} = \frac{p}{p} = 1$$