= Asymmetrically almost-subadditive sequence
{title2=$x_{m+n}\leq x_m+x_n+\alpha_n,\quad\alpha_n=o(n)$}
If the real error depends only on the second index and $\alpha_n/n\to0$, the normalized sequence has a limit in $[-\infty,\infty)$. For fixed $k$, write $n=qk+r$ with $1\leq r\leq k$ and iterate to obtain $x_n\leq x_r+q(x_k+\alpha_k)$. It follows that $\limsup x_n/n\leq(x_k+\alpha_k)/k$ 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.
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