So i am an undergrad student and this question was asked in an assignment
Examine if $\{(x, y)\in \Bbb R^2 \mid 2x+3y≤6,2x+3y≥6, x≥0, y≥0\}$ is a convex set
After solving the constraints, we come to the conclusion that the set is basically a line segment joining points (3, 0) and (0, 2) and since a line segment is convex the given set is a convex set. Or we can also use the theorem that intersection of two convex sets is a convex set. Now the two constraints in the set individually give us 2 half planes and since half planes are a convex set then their intersection will also be a convex set. Are these above reasons sufficient to prove that the given set is convex?
Intersection of convex set is convex is indeed the result that you need.
Each of the constraints is a half space which is convex, hence the intersection must be convex.
In general, a polyhedral which are intersection of half spaces are convex.