Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 133 / 3 / a / Solution

For each generator $t\in T$, let $|t{|}_S$ be its length in the generating set $S$, and putA shortest $T$-word for $h_1^{-1}{h}_2$ has $d_T(h_1,h_2)$ letters. Replacing each letter by an $S$-word of length at most $C$ givesThus inclusion of any finitely generated subgroup is Lipschitz continuous for the corresponding word metrics.