Let \(\left\{\mathbf{X}_{n}ight\}_{n=1}^{\infty}\) be a sequence of random vectors that converge almost certainly to a random vector \(\mathbf{X}\) as \(n ightarrow \infty\). Prove that for every \(\varepsilon>0\) it follows that

\[\lim _{n ightarrow \infty} P\left(\left\|\mathbf{X}_{m}-\mathbf{X}ight\|<\varepsilon \text { for all } m \geq night)=1\]

Prove the converse as well.