  Consider a population whose size changes stepwise by its members reproducing
or 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 size
changes, 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 class
of population sizes. We make no assumptions about the time durations between
the successive changes. In the real world, or more specific models, those may
be of varying length, depending upon individual life span distributions and
their interdependencies, the age-distribution at hand and intervening
circumstances. Changes may have quite varying distributions. The basic
assumption is that there is a {\em carrying capacity}, i.e. a non-negative
number $K$ such that the conditional expectation of the change, given the
complete past history, is non-positive whenever the population exceeds the
carrying capacity. Further, to avoid unnecessary technicalities, we assume that
the change $C_n$ equals -1 (one individual dying) with a conditional (given the
past) probability uniformly bounded away from 0. It is a simple and not very
restrictive way to avoid parity phenomena, it is related to irreducibility in
Markov settings. The straightforward, but in contents and implications
far-reaching, consequence is that all such populations must die out.
Mathematically, it follows by a submartingale convergence property and positive
probability of reaching the absorbing extinction state.
