Let \(f_1,f_2,\ldots \) be a sequence of real-valued convex functions on \(\mathbb R\). Define their upper-limit via \[ \overline f(x)=\limsup _{n\to \infty } f_n(x); \] and the lower-limit via \[ \underline f(x)=\liminf _{n\to \infty } f_n(x). \] Prove or find a counterexample to each of the following two statements:
(i) \(\overline f\) is convex.
(ii) \(\underline f\) is convex.