Finite index in finite double coset index implies finite double coset index

Statement with symbols
Suppose $$H \le K \le G$$ are groups such that $$H$$ has finite index $$r$$ in $$K$$, and $$K$$ has finite fact about::double coset index $$s$$ in $$G$$: there are $$s$$ double cosets of $$K$$ in $$G$$. Then, $$H$$ has finite double coset index in $$G$$, and the double coset index of $$H$$ in $$G$$ is bounded by:

$$r + r^2(s - 1)$$.

Other facts about index and double coset index

 * Index is multiplicative: If $$H \le K \le G$$, then $$[G:K][K:H] = [G:H]$$ in the sense of possibly infinite cardinals. In particular, if $$K$$ has finite index in $$G$$ and $$H$$ has finite index in $$K$$, then $$H$$ has finite index in $$G$$.
 * Finite double coset index is not transitive: We can have a situation of groups $$H \le K \le G$$ such that $$H$$ has finite double coset index in $$K$$ and $$K$$ has finite double coset index in $$G$$, but $$H$$ does not have finite double coset index in $$G$$.

Tightness of the result
For $$G$$ equal to the dihedral group of order $$2p$$ for an odd prime $$p$$, $$K$$ of order two, and $$H$$ trivial, the bound is tight. More generally, the bound is tight when $$G$$ is a Frobenius group, $$K$$ is a Frobenius complement in $$G$$, and $$H$$ is the trivial subgroup.