Below we have the definition of a convex set.I want to prove that sum of convex sets is a convex set.using definition bellow i take two points from each set $x'_1,x'_2\in S1$ and $x''_1, x''_2 \in S2$. For each set we have the following expression
$$\lambda'x'_1+(1-\lambda')x'_2 \in S1 \\ \lambda''x''_1+(1-\lambda'')x''_2 \in S2 $$
If i want to show that their sum is convex too,i need to arrange it in the $\lambda x+(1-\lambda)x \in S1$ structure structure. How can i do it? Thanks
Update:I have seen solutions to these proof as shown bellow. the main problem with this proof is that they share the same lambda unlike what i tried to have separated lambdas. Why is that?