Let $A, G \subset C ([a, b])$, $G = \{g_1, g_2, ..., g_m\}$ (finite set). Prove that if: i) $A || .. ||$ $\infty$-bounded then $A \cup G$ too. ii) $A$ equicontinuous in $x_o$ then $A \cup G$ also.
for i) since $G$ is finite it has a max and a min element. I tried the triangle inequality taking the distance of any two element of $A$ and $G$.
ii) Don't know how to proceed.
Show from the definitions:
A union of two $\infty$-bounded sets is $\infty$-bounded.
A finite set is $\infty$-bounded.
Show the same two facts for equicontinuous sets.
For the finite case use that a single continuous function on $[a,b]$ is bounded and uniformly continuous.