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Let $f^{(r)}(n;s,k)$ be the maximum number of edges of an $r$-uniformhypergraph on~$n$ vertices not containing a subgraph with $k$~edges and at most$s$~vertices. In 1973, Brown, Erd\H{o}s and S\'os conjectured that the limit$$\lim_{n\to \infty} n^{-2} f^{(3)}(n;k+2,k)$$ exists for all positive integers$k\ge 2$. They proved this for $k=2$. In 2019, Glock proved this for $k=3$ anddetermined the limit. Quite recently, Glock, Joos, Kim, K\"{u}hn, Lichev andPikhurko proved this for $k=4$ and determined the limit; we combine their workwith a new reduction to fully resolve the conjecture by proving that indeed thelimit exists for all positive integers $k\ge 2$.

The limit in the $(k+2, k)$-Problem of Brown, Erd\H{o}s and S\'os exists for all $k\geq 2$