Ramsey number recursion ($R(k,\ell )\le R(k-1,\ell )+R(k,\ell -1)$)

Let $A=R(k-1,\ell )$ and $B=R(k,\ell -1)$. From one vertex of any red-blue colored $K_{A+B}$, either at least $A$ incident edges are red or at least $B$ are blue. In the first case, apply the definition of the Ramsey number to its red neighbors: a red $K_{k-1}$ extends with the chosen vertex, and a blue $K_{\ell }$ is already enough. The blue case is analogous. This proves the recursion, with boundary values $R(k,2)=k$ and $R(2,\ell )=\ell$.