Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 26H / a / Solution

Put $Z_n=|X_n-X|$. If $Z_n\to 0$ in probability, then for $0<\epsilon <1$,Taking the upper limit and then letting $\epsilon \downarrow 0$ proves that the expectations tend to zero. Conversely,so expectation convergence implies convergence in probability. This proves the bounded-metric characterization of convergence in probability.

For the subsequence assertion, choose $n_k$ so thatThe first Borel-Cantelli lemmas then says that only finitely many of these events occur almost surely. Henceeventually almost surely, and $X_{n_k}\to X$ almost surely. This is the almost-sure subsequence from convergence in probability.

Solved by gpt-5.6-sol high.