This is an interesting question regarding Maple, although this question appears to be pure math, I recommend these two resources:

• http://mathonline.wikidot.com/cauchy-sequences-of-real-numbers
• https://brilliant.org/wiki/cauchy-sequences/

Also, there is a way to play and verify whereas sequence is Cauchy or not, I will give you some examples using Maple:

Example 1:

Show that the sequence  is a Cauchy Sequence, ok, first you may the sequence behave by generating the sequence using seq command, but first, define a function

Next, generate a sequence with: ; let say you want to see how the sequence behaves using the first 50 items from the sequence. By merely observing the behavior of the sequence you may notice that the sequence tends to converge to 0, whch by definition, a convergence sequence is also a Cauchy Sequence– you may verify the links above.

Then, using the definition by choosing an n which belongs to N choose an n sufficiently big to verify the Cauchy sequence definition as follows:

finally, the result of that evaluation compare with an epsilon, let say epsilon = 0.05 and if your results < than epsilon then empirically the sequence is a Cauchy Sequence. You may do this with Maple and that's it.

Example 2:

Show that n is a Cauchy Sequence. Repeat these steps:

For this example, you will find that, this is not a Cauchy Sequence.

Example 3:

An example more interesting in R^2, the Same strategy, follow the next steps:

And that's it.

I hope this helps to you, but I see this question is five years old, probably help others too :P using our Beloved Maple :*

Very Best Regards