Question 1: Is my proof that $\operatorname{Spec}k[T_1,T_2,\dots]-V((T_1,T_2,\dots))$ isn't quasi-compact correct?
Here is my attempt: Let $X$ denote said scheme. Consider the sequence of ideals $$(0)\subset(T_1)\subset(T_1,T_2)\subset\cdots$$ to which there is a sequence of spaces $V(0) = X\supset V(T_1)\supset V(T_1,T_2)\supset\cdots$ and their intersection is $$\bigcap_{n\geq 0}V(T_1,\dots,T_n) = V(T_1,T_2,\dots)=\varnothing$$ and there is evidently no finite subcollections of the $V(T_1,\dots,T_n)$ whose intersection is empty, hence $X$ fails FIP on closed sets.
Question 2: In Brian Conrad's notes on the Proj construction, he says that the scheme $\operatorname{Spec}k[T_1,T_2,\dots]-\{(0)\}$ is not quasi-compact, but I'm unable to convince myself of this. Is there an easy proof along the lines above?