A
sequence x (i.e. a
function x
mapping N to a
metric space S) is said to
converge if there exists an
element L of S and a
gauge function N:(0,
infinity)->
N such that for every
positive real number e, d(x(n),L) < e whenever n > N(e). If this is the case, L is called the
limit of the sequence x.
Theorem
A real sequence x:N->R converges if and only if there exists a guage function N:(0,infinity)->N such that for every positive real number e, d(x(n),x(m)) < e whenever n > N(e) and m > N(e).
This property is called the Cauchy criterion, and a sequence satisfying it is said to be a Cauchy sequence. The strength of this theorem is that it determines a necessary and sufficient criterion for convergence. However, it does not provide a method for determining the limit of a sequence; this is a much more difficult problem, and cannot be solved for a general sequence.
A
function f:D->
R is sait to
converge at a
point t in D provided that for some
real number L and some
guage function d:(0,
infinity)->(0,infinity), one has:
For all e > 0, for all s in D, 0 < |s-t| < d(e) implies |f(s) - L| < e.
Then, L is said to be the
limit of f at t.