Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 12D / b / iii / Solution

For $k\ge 1$, let $E_k$ be the set of numbers whose first digit $5$ occurs in position $k$. The first $k-1$ digits each have nine allowed values, while the $k$th is fixed, so $E_k$ is a union of decimal intervals of total lengthThe set $E_{\infty }$ of numbers having no digit $5$ can, after $N$ places, be covered by intervals of total length $(9/10{)}^N$; hence it has length zero.

For each fixed truncation level, the function is constant on finitely many decimal cylinders apart from the remaining set, whose covering length can be made arbitrarily small. The Riemann integrability criterion therefore shows that every truncation is Riemann integrable.

For integer $N$,The final term tends to zero, and monotonicity handles noninteger truncation levels. Therefore the first occurrence of a decimal digit calculation gives

Solved by gpt-5.6-sol high.