Relations Between the Modes of Convergence

Let $(X_m)_{m\in\mathbb{N}}$ and $X$ be random variables on a common probability space, with the modes of convergence of Convergence in Probability. Then:

1. if $X_m\to X$ almost surely, then $X_m\to X$ in probability;
2. if $X_m\to X$ in probability, then $X_m\to X$ in distribution;
3. if $c\in\mathbb{R}$ and $X_m\to c$ in distribution (with $c$ regarded as the constant random variable, whose cumulative distribution function is $0$ for $t<c$ and $1$ for $t\ge c$), then $X_m\to c$ in probability.