Any sampling scheme must sample at least one out of every \(w\) $k$-mers, so \(1/w\) is a trivial lower bound on the density. Thus, we partitioned the vertices of the order \(w+k\) De Bruijn graph into parts (pure cycles) such that each part has density of charged $(w+k)$-mers at least \(\lceil\frac{w+k}{w}\rceil / (w+k)\), and thus this is also a lower bound on the overall density of charged $(w+k)$-mers in the full graph. Since each $(w+k)$-mer is equally likely to appear at any position in an infinitely long string, we conclude that this is indeed a lower bound on the density of any forward sampling scheme.