Past exam of the computer science course of the University of Oxford / 2015 / Probability and computing / 2 / c / Solution

The printed bound also fails for the actual expectations. In the two-bin counterexample from part (b), at $j=5$ the true value and proposed bound areThe supplied premise $x_2\le n/2$ does hold in this example: $x_2=1/2\le 1$. Thus the problem is the invalid recurrence in part (b), rather than failure of the supplied starting estimate.^[ref]

The intended mathematical induction is valid conditionally on the recurrence. For a deterministic nonnegative sequence $y_j$ satisfying $y_2\le n/2$ and $y_{j+1}\le y_j^2/n$, the base case is $y_2\le n{2}^{-1}$. If $y_j\le n{2}^{-2^{j-2}}$, thenThis proves the proposed form for $y_j$ but cannot identify $y_j$ with $\mathbb{E}{B}_j$.