Examples grounded in the definition:

• converges to : given , the Axiom of Archimedes supplies , and then for all .

• The sequence defined by for even and for odd converges to by the same choice of , treating the two parities separately.

• converges to , since .

• The alternating sequence diverges: with , no tail can stay within of any proposed limit, since . Divergence does not require the terms to tend to infinity.