Claim: The space $C_c$ of real valued continuous, compactly supported functions is not complete
Solution:
I don't really know how to prove this claim. Nonetheless, I had some Ideas: I was able to prove that $C_c$ is not closed. Also we know that every complete subspace of a metric space is closed. But I cannot conclude. Can someone help me?
Thank you
Best Answer
You don't mention the norm for completion? but with the sup norm you may take a sequence $f_n\in C_c$ converging to e.g. $1/(1+x^2)$ uniformly (just truncate in a continuous way, not by indicator function). Then $f_n$ is Cauchy but does not converge in $C_c$.