# Prove that the intersection of a number of finite convex sets is also a convex set

## Homework Statement

Prove that the intersection of a number of finite convex sets is also a convex set

## Homework Equations

I have a set is convex if there exists x, y in the convex S then

f(ax + (1-a)y< af(x) + (1-a)y

where 0<a<1

## The Attempt at a Solution

i can prove that
f(ax + (1-a)y) < f(x) given that x is a global minimizer

then i guess that i could find another arbritary point close to x , x_1, x_2 and add their given function satisfying the convex condition to get

Sum f(axi + (1-a)y) < Sumf(xi) where i= 1, 2,...n

any help would be appreciated

Your definition of "convex set" is wrong; there is no function involved. A set $$S$$ (in some real vector space $$V$$) is convex if, whenever $$x, y \in S$$ and $$0 \leq a \leq 1$$, then also $$ax + (1 - a)y \in S$$.