Question: (a) Let \(U_{1}, U_{2}, \ldots\) be independent identically distributed random variables with a U[0,1] distribution. Call the \(n^{\text {th }}\) realisation a record if \(U_{n}\) is bigger than any previous value. Prove that there will be infinitely many records. [Hint: use the Borel-Cantelli Lemmas]

(b) Prove that if \(X_{n} \rightarrow X\) almost surely, and \(Y_{n} \rightarrow Y\) almost surely, then \(X_{n} Y_{n} \rightarrow\) \(XY\) almost surely.

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