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 <monotone bounded sequence>[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 <convergent sequence>[limit].

Solution

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

 Solution (source code)