5. Let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space, and $A_1, A_2, \dots$ be an increasing sequence of events; that is, $A_1 \subseteq A_2 \subseteq \dots$ such that $\mathbb{P}(A_1) = \mathbb{P}(A_2) = \dots = p > 0$. Does the sequence of events converge in probability to the event $A = \bigcup_{n=1}^{\infty} A_n$? Prove or disprove this. [2]
Hint: For a sequence of events, convergence in probability can be written as $\lim_{n \to \infty} \mathbb{P}(A \triangle A_n) = 0$, where $\triangle$ is the symmetric difference.
