We have a sequence of continuous functions $f_n : [0,\infty) \to \Bbb R$ such that $f_n \to f$ uniformly. Also each $f_n$ has a zero. But $f$ does not have any zero. Then what is the example of such a sequence of functions and their limit function.
I am trying to think about a sequence of functions such that their limit function is $f(x)=1 \quad \forall x \in [0,\infty)$. However, I am really stuck on this for a long time and completely clueless after trying many functions $f_n$.
Here is the link where I proved (after getting help) that if the domain is $[a,b]$ then $f$ will always have a zero because of sequential compactness of $[a,b]$. In case of $[0,\infty)$ it is not possible.