How can I show that the space of continuous real valued functions on R with compact support in the usual sup norm metric is not complete ? I know that this result can be proved by using the fact that the given space is dense in the space of all continuous functions that vanish at infinity which is complete , but I want a bit easier proof of this as I have not studied measure theory . I was trying to solve it directly from the definition of completeness.

Thanks for any help.


Take a particular example of a continuous function that goes to $0$ at $\pm \infty$, and a sequence of continuous functions of compact support that converges uniformly to it. This is a Cauchy sequence ...

  • $\begingroup$ Hello @Robert ;I took the function to be $f(x)=\dfrac{1}{1+x^2}$ but how to define the sequence that converges to $f$ ;Can you please help me $\endgroup$
    – Learnmore
    Oct 14 '16 at 14:02
  • 1
    $\begingroup$ For example, $f_n(x) = \dfrac{1-(x/n)^2}{1+x^2}$ for $-n \le x \le n$, $0$ otherwise. $\endgroup$ Oct 14 '16 at 15:25
  • $\begingroup$ @jaggu That's precisely why it should be $[-n,n]$. $\endgroup$ Dec 28 '16 at 16:57
  • $\begingroup$ @RobertIsrael......... oops, continuity, sorry $\endgroup$
    – jaggu
    Dec 28 '16 at 20:59
  • $\begingroup$ @RobertIsrael, do you know how to find the completion of this metric space in question? $\endgroup$
    – User1618
    Dec 7 '18 at 16:34

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