I have the following question;
Let $E \subseteq \mathbb{R}^n$ be a convex subset, and $ f: E \rightarrow \mathbb{R}$ be a convex function. Show that for any $c \in \mathbb{R}$, the set $A=\{x \in E \; | \;f(x) \leq c\}$ is a convex subset of $\mathbb{R}^n$. Show furthermore that if $\{f_i\}_{i\in I}$ is any collection of convex functions on a convex subset $E \subseteq \mathbb{R}^n$, and $\{c_i\}_{i\in I}$ is a collection of real numbers, then the set $B=\{x \in E \;|\; f_i(x) \leq c_i, \forall i \in I \}$ is a convex subset of $\mathbb{R}^n$.
I am confident in the first part of the proof; if $x, y \in A$ then $$f((1-t)x+ty) \leq (1-t)f(x)+tf(y) \leq (1-t)c+tc=c$$ So $(1-t)x+ty \in A$ and $A$ is convex as required.
I believe the second part of the proof is analogous. I have;
If $x,y \in B$ then for any $i$,
$$f_i((1-t)x+ty) \leq (1-t)f_i(x)+tf_i(y) \leq (1-t)c_i+tc_i=c_i$$
So $(1-t)x+ty \in B$ and $B$ is convex. I don't feel I would be asked to do essentially the same proof twice, am I missing something?