Assume that values of
can always be restricted to arbitrary open ranges around
by restricting input values
to open subdomains around
(but excluding
itself). We would like to verify that
as
So assume that input sequence
from within the domain of
(but, as always,
for all
). We need to verify that the corresponding output sequence
converges to
So consider an open range
around
From our assumption, it’s possible to specify a corresponding open domain
around
so that values of
are always within
for
within
(but ignoring
). As
and
lies within
there is a tail
that is completely contained in
But within this subdomain all output values of
lie within range
In particular, the output values
all lie within
for the terms in the tail is contained in
That is, the corresponding tail
of the output sequence is entirely contained within
Since such a tail of the output sequence can always be determined for every open range
around
we conclude that