= Almost-additive partition-function limit
{title2=$|a_{n+m}-a_n-a_m|\leq D\ \Longrightarrow\ \lim a_n/n\text{ exists}$}
The sequences $a_n+D$ and $a_n-D$ are respectively subadditive and superadditive. Apply <Fekete's lemma> to the first and its negative to the second; their normalized difference is $2D/n$, so their limits coincide. For full-shift weighted orbit sums, summable variations of the potential bound the energy change at a concatenation boundary, giving this estimate for the logarithm of the <partition function>. A fixed initial symbol or fixed remote tail changes only a bounded normalization and not the limit.
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