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Is the followimg statement is True/false

The Space of all continiuos real valued functions with compact support with supnorm metric is complete . (True/false)

i have found the answer here : are they complete metric spaces?

Now my confusion is that i didn't understand the answer How can we contradicts this function ?

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    $\begingroup$ Is this a homework problem? $\endgroup$ Aug 5 '19 at 23:53
  • $\begingroup$ @KaviRamaMurthy No sir , Im preparing For NBHM exam it is a NBHM question $\endgroup$
    – jasmine
    Aug 5 '19 at 23:55
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Hints: Each $f_n$ has compact support since $x^{2}>n-1$ implies $f_n(x)=0$, Verify that $f_n(x) \to f(x) \equiv \frac 1 {1+x^{2}}$ uniformly on $\mathbb R$. Conclude that $\{f_n\}$ is Cauchy in the given space. Suppose it converges to some $g$ in the given space. Then $f_n \to f$ and $f_n \to g$ pointwise. Hence $f(x)=g(x)$ for all $x$. But $g$ has compact support and $f$ doesn't. This completes the proof.

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  • $\begingroup$ Im not getting How $ f$ does not have compact support i mean u have already take that each $f_n $ has compact support $\endgroup$
    – jasmine
    Aug 5 '19 at 23:51
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    $\begingroup$ $f(x)=\frac 1 {1+x^{2}} \neq 0$ for any $x$. $\endgroup$ Aug 5 '19 at 23:53
  • $\begingroup$ Got its sir thanks u sir @Kavi rama sir $\endgroup$
    – jasmine
    Aug 5 '19 at 23:55

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