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Consider a population whose size changes stepwise by its members reproducingor dying (disappearing), but is otherwise quite general. Denote the initial(non-random) size by $Z_0$ and the size of the $n$th change by $C_n$, $n= 1, 2,\ldots$. Population sizes hence develop successively as $Z_1=Z_0+C_1,\Z_2=Z_1+C_2$ and so on, indefinitely or until there are no further sizechanges, due to extinction. Extinction is thus assumed final, so that $Z_n=0$implies that $Z_{n+1}=0$, without there being any other finite absorbing classof population sizes. We make no assumptions about the time durations betweenthe successive changes. In the real world, or more specific models, those maybe of varying length, depending upon individual life span distributions andtheir interdependencies, the age-distribution at hand and interveningcircumstances. Changes may have quite varying distributions. The basicassumption is that there is a {\em carrying capacity}, i.e. a non-negativenumber $K$ such that the conditional expectation of the change, given thecomplete past history, is non-positive whenever the population exceeds thecarrying capacity. Further, to avoid unnecessary technicalities, we assume thatthe change $C_n$ equals -1 (one individual dying) with a conditional (given thepast) probability uniformly bounded away from 0. It is a simple and not veryrestrictive way to avoid parity phenomena, it is related to irreducibility inMarkov settings. The straightforward, but in contents and implicationsfar-reaching, consequence is that all such populations must die out.Mathematically, it follows by a submartingale convergence property and positiveprobability of reaching the absorbing extinction state.

Populations in environments with a soft carrying capacity are eventually extinct