My proof homework states:
As per A set C of real numbers is convex if and only if for all elements in x, y ∈ C and for every real numbers t with 0 ≤ t ≤ 1, tx + (1 − t)y ∈ C. Suppose a, b ∈ R. Show that the set C = {x ∈ R | ax ≤ b} is convex.
For the solution, it states: "Thus we are given ax ≤ b and ay ≤ b."
I don't know if I missed the axiom on this, but I can't find the proof online as to why this is. Can someone help?
Take $x,y\in C$. Then $ax\leq b$ and $ay\leq b$. Thus for each $t\in[0,1]$, $$a(tx+(1-t)y) = tax + (1-t)ay \leq tb + (1-t)b = b.$$