Problems:
For each positive integer k define
a.
b.
c.
for . Is the sequence a Cauchy Sequence in the metric space
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By definition, a sequence is Cauchy sequence in if and only if there is an
such that
for all x in [a, b]
For a I said that is only Cauchy for when since the distance between the get smaller and smaller together. Thus this sequence is a Cauchy sequence.
For b, , I said that this is not Cauchy as it is distance is getting larger and larger since the sequence of function grows exponentially. So as k and l goes to infinity, does not go to zero.
For c, , I said that this is Cauchy with a similar argument with part (a).
I have no idea how to put this in a formal proof though, but the problem seems like it only wants a true or false type of answer, I think.
Thank you for your time.