Discussion
Suppose X is a real random variable on a probability space \((\Omega, A, P)\) and \(\{A_{n}, n \ge 1\}\) is a sequence of events in A such that :
\[ A_{n} \to \Omega \text{ as } n \to \infty. \]
Which of the following is true ?
Answer: C
\(X I_{A_{n}} \xrightarrow{P} X\).
\(A_{n} \to \Omega\) means that every outcome eventually belongs to \(A_{n}\) (for the limit set to be \(\Omega\), \(\liminf A_{n} = \Omega\)). So for every \(\omega\), \(I_{A_{n}}(\omega) = 1\) for all large \(n\), and \(X I_{A_{n}}(\omega) = X(\omega)\) from some point on. That is almost sure, indeed pointwise, convergence to X, which implies convergence in probability.
In probability terms, \(P(|X I_{A_{n}} - X| \gt \varepsilon) \le P(A_{n}^{c}) \to 0\) by continuity of P.
The limit is not 0 (in probability or in law) unless X itself is 0, and it is not the constant 1 unless X is 1.
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