Is the set of all real valued continuous functions on $\mathbb R$ with compact support complete?
the metric is the usual sup-norm metric
how to do it any hints .I cant prove it neither can find a counter example
previous answers dont satisfy me
[Math] Is the set of all real valued continuous functions on $\mathbb R$ with compact support complete
real-analysis
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Best Answer
Let $\phi(x)$ be a nonzero continuous function with support inside $[0, 1]$. Then let
$$f_1(x) = \phi(x), \ \ \ f_2 = \phi(x) + \frac{1}{2} \phi(x-1), \cdots, f_n = \sum_{i=0}^{n-1} \frac 1{i+1}\phi(x-i)$$
Then we have $f_i \in C_c(\mathbb R)$ and is a Cauchy sequence. The sequence $\{f_i\}$ converges uniformly to a continuous function
$$f = \sum_{i=0}^\infty \frac 1{i+1} \phi(x-i)$$
but $f$ is not of compact support. Thus $C_c(\mathbb R)$ is not complete with respect to the sup norm.