Probability of events tends to zero implies probabilty of liminf is zero

Question

Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of events such that $\mathbb{P}[A_n] \to 0$. Does this imply that $\mathbb{P}[\lim\inf_n A_n] = 0$? How would one establish this implication if its true, or perhaps a counter-example if not?

Answer 1

Here, lim inf$_n$ $ A_n = \cup_{m=1}^{\infty} \cap_{n=1}^{m} A_n$

$\Rightarrow$ lim inf$_n$ $ A_n \subseteq A_m $, $\forall$ $ m\geq n_0 $, for some $n_0\in \mathbb{N}$.

Hence, by monotonicity of probability measure,

$\mathbb{P}$ (lim inf$_n$ $ A_n$) $\leq \mathbb{P}(A_m)$, $\forall$ $ m\geq n_0 $, for some $n_0\in \mathbb{N}$.

Now taking $m\to \infty$ and using $0\leq\mathbb{P}(A)\leq1$, in above statement, desired result is obtatined.