If $X_n \stackrel{p, quickly}{\to} X$, then $X_n \to X$.

Question

Probability with Martingales:

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1. Without using hint, can I just do something like this: https://math.stackexchange.com/a/1538503/140308 ?

2. With using hint:

By continuity of probability, I think we have

$$P(\bigcup_{k=1}^{\infty} B_k) = \lim_{k \to \infty} P(B_k)$$

where $B_k := \limsup_n A_{n,k}$

where $A_{n,k} := [|X_n - X| > \frac{1}{k}]$

By replacing, in our hypothesis, $\varepsilon$ with $\frac{1}{k}$ and by BCL1, I think we have

$$P(B_k) = 0$$

Hence

$$P(\bigcup_{k=1}^{\infty} B_k) = \lim_{k \to \infty} P(B_k) = \lim_{k \to \infty} 0 = 0$$

Is that right?