Answer by Roland Bacher for What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?

This fly can also be killed using the Leech-lattice as a sledgehammer:

The existence of the Leech lattice (an even unimodular (integral) lattice of dimension 24 and (squared) minimum 4) implies that the 24-dimensional unit ball $B_{24}$ has volume $V<1$. This implies that the $24n$-dimensional unit ball $B_{24n}$ (contained in the product of $n$ copies of $B_{24}$) has volume at most $V^n$ which decays exponentially fast to $0$ for $n\rightarrow \infty$. The general case can be handledby observing that the $24n+k$-dimensional unit ball $B_{24n+k}$ with $k\in\{0,\ldots,23\}$ is contained in $B_{24n}\times[-1,1]^k$ of volume at most $2^{23} V^n$.