Subgroups of finitely generated nilpotent groups are finitely generated

Put $K_i=K\cap {\gamma }_i(G)$. The homomorphism $K_i\to {\gamma }_i\mathrm{/}{\gamma }_{i+1}$ has kernel $K_{i+1}$. Its image is a subgroup of a finitely generated abelian group, so has a finite generating list. Thus $K_i\mathrm{/}{K}_{i+1}$ is finitely generated: explicitly its isomorphism to that image sends $x{K}_{i+1}$ to $x{\gamma }_{i+1}$, with injectivity given by the kernel calculation.

Start with $K_{c+1}=1$. Lift a finite generating list of $K_i\mathrm{/}{K}_{i+1}$ to $K_i$. For $x\in K_i$, a word $w$ in these lifts has coset $x{K}_{i+1}$, so $w^{-1}x\in K_{i+1}$. Consequently those lifts together with generators of $K_{i+1}$ generate $K_i$. There are finitely many layers, hence this gives finite generators of $K_1=K$. Empty factor lists require no lift, and $c=0$ gives $K=1$.