Editing Probability axioms (section)

==== ''Proof of monotonicity'' ====
Source:<ref name=":1" />

In order to verify the monotonicity property, we set <math>E_1=A</math> and <math>E_2=B\setminus A</math>, where <math>A\subseteq B</math> and <math>E_i=\varnothing</math> for <math>i\geq 3</math>. From the properties of the [[empty set]] (<math>\varnothing</math>), it is easy to see that the sets <math>E_i</math> are pairwise disjoint and <math>E_1\cup E_2\cup\cdots=B</math>. Hence, we obtain from the third axiom that

:<math>P(A)+P(B\setminus A)+\sum_{i=3}^\infty P(E_i)=P(B).</math>

Since, by the first axiom, the left-hand side of this equation is a series of non-negative numbers, and since it converges to <math>P(B)</math> which is finite, we obtain both <math>P(A)\leq P(B)</math> and <math>P(\varnothing)=0</math>.