Relations among notions of convergence Let $\\{A_n\\}_{n \in \mathbb{N}}$ be a sequence of real numbers such that $\lim_{n \rightarrow \infty}A_n=0$. Does this imply that $plim_{n\rightarrow \infty}A

Relations among notions of convergence Let $\\{A_n\\}_{n \in \mathbb{N}}$ be a sequence of real numbers such that $\lim_{n \rightarrow \infty}A_n=0$. Does this imply that $plim_{n\rightarrow \infty}A_n=0$, where $plim$ is the probability limit?

Using Markov's inequality:

$$ P( |A_n-0|> \epsilon) \leq \frac{E(|A_n|)}{\epsilon}=\frac{|A_n|}{\epsilon} $$ Letting $n\rightarrow \infty$ on both sides proves convergence in probability.