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

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 ?

• Is this a homework problem? Aug 5 '19 at 23:53
• @KaviRamaMurthy No sir , Im preparing For NBHM exam it is a NBHM question Aug 5 '19 at 23:55

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.

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