Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 2E / iii / Solution

Part (i) also shows that the odd subsequence is increasing and the even subsequence is decreasing. Each odd term is below the adjacent even term, so both are bounded monotone sequences and therefore have limits. Part (ii) says their difference tends to zero, so those limits coincide. Interlacing the two subsequences proves that $(a_n)$ has that common limit.