Asymmetrically almost-subadditive sequence ($x_{m+n}\le x_m+x_n+{\alpha }_n,{\alpha }_n=o(n)$)

If the real error depends only on the second index and ${\alpha }_n/n\to 0$, the normalized sequence has a limit in $[-\infty ,\infty )$. For fixed $k$, write $n=qk+r$ with $1\le r\le k$ and iterate to obtain $x_n\le x_r+q(x_k+{\alpha }_k)$. It follows that $\mathrm{limsup}{x}_n/n\le (x_k+{\alpha }_k)/k$ for every $k$; a subsequence attaining the lower limit proves convergence. The limit is ${\inf }_k(x_k+{\alpha }_k)/k$, even when the errors can be negative.