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 ?
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.