Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 23H / e / Solution

For integer $k\ge 1$, define $f_k(n)=f(k,n)$. The assumed monotonicity in $t$ givesfor every $n$. Apply the monotone convergence theorem on $\mathbb{N}$ with counting measure:The function $t\mapsto {\sum }_{n}f(t,n)$ is itself nondecreasing, and the positive integers are cofinal in $\mathbb{R}$ as $t\to \infty$, so its limit over real $t$ equals its limit over integer $k$. Therefore

Without monotonicity, takeFor each fixed $n$, $f(t,n)\to 0$, so the sum of the pointwise limits is zero. At every $t\ge 1$, however, exactly one term is one, so ${\sum }_{n}f(t,n)=1$ and its limit is one. This moving-spike obstruction to interchanging a limit and an infinite sum gives the required counterexample.