To prove a sequence is Cauchy from the definition, an opening line of proof may go something like as follows.
Define the sequence
converges by showing
Ex 1 Comment
can be verified directly from the definition of convergence, but that is not permitted by this prompt. The point here is to gain practice in a new technique.
be given. It suffices to show there is a natural number
, for all
Observe, for all indices
where the first inequality is an application of the triangle inequality and the second follows from the fact
. By the Archimedean Property of
, there is a natural number
, and so
This shows the desired inequality holds, taking
, by which we conclude