Is the following Proof Correct?
Proposition. Consider an experiment whose sample space consists of a countably infinite number of points. Show that not all points can be equally likely.
Proof. Let $\mathcal{S} = \{\alpha_1,\alpha_2,\alpha_3,\dots\}$ be the sample space for the experiment in question and assume on the contrary that $\mathbf{P}(\alpha_i) = p,\forall i\in \mathcal{S}$ where $p\in [0,1]$. If $p=0$ an appeal to the second and third axiom of probability implies that $1 = \mathbf{P}(\mathcal{S}) = \mathbf{P}(\cup_{i=1}^{\infty}\{\alpha_i\}) = \sum_{i=1}^{\infty}\mathbf{P}(\{\alpha_i\}) = 0$ resulting in a contradiction, similarly in the event $p\in(0,1]$ an appeal to the third axiom indicates that $\mathbf{P}(\cup_{i=1}^{\lceil\frac{1}{p}\rceil+1}\{\alpha_i\}) = \sum_{i=1}^{\lceil\frac{1}{p}\rceil+1}\mathbf{P}(\{\alpha_i\}) = (\lceil\frac{1}{p}\rceil+1)\cdot p\ge 1$ contradicting the first axiom. We may therefore conclude that our original assumption was incorrect and that the probability of all events in $\mathcal{S}$ is not equally likely.
$\blacksquare$
$PS:$ A Follow up question concerning the above problem also asked whethere all points have a positive probability of occurring?
I believe that the answer is no for the same reasons as those presented in the above proof for case $p\neq 0$.