Completeness of set

Asked Mar 18 '14 at 01:43

Active Mar 18 '14 at 01:46

Viewed 40 times

Show that the set of bounded continuous functions is complete in the sup-norm.

My work: we want every Cauchy sequence to have a limit in the same set. But I don't understand what we can start with. If we take some Cauchy sequence $||f_n - f_m|| \leq \epsilon$, but we still have no information about its convergence...

asked Mar 18 '14 at 01:43

user136153

I saw the question, but it starts the solution at a level I don't understand. – user136153 Mar 18 '14 at 01:47
For example, "Since for fixed $x$, $f_n(x)$ is a Cauchy sequence in $\mathbb R$" - i don't see why? – user136153 Mar 18 '14 at 01:50
That is a claim in the first proof. – Mar 18 '14 at 01:52
Is this claim based on a theorem? Where does it come from? – user136153 Mar 18 '14 at 01:52
That is an assumption which is used to prove and simplify the statement. – Mar 18 '14 at 01:53

Completeness of set

0 Answers0