Proof for Lemma about convex hull I have to prove a Lemma: "For the set B of all convex combinations of arbitrary finite number of points from set A, $co (A)=B$"  
I started by showing $B\subset co(A)$ first. 
$B$ contains all convex combinations of arbitrary finite number of points from A.
Let $x=\alpha_1 x_1 +...+\alpha_n x_n$ be convex combination of $x_1,...x_n\in A$
$x\in B$
Let $C$ be any convex set that contains A.
Now I know that $x_1,...,x_n\in C$ and, since $C$ is convex, it contains all convex combinations of arbitrary finite number of its points (and points from A), so $x\in C$.
Thus, $B\subset C$
I also know that $C=co(C)$ because $C$ is convex.
Also, $A\subset C \rightarrow co(A)\subset co(C) \rightarrow co(A)\subset C$  
So I have $B\subset C$ and $co(A)\subset C$.
How can I conclude from this that $B\subset co(A)$?  
What about another direction, $co(A)\subset B$? Just to prove that $B$ is convex?

We start with the set $B$ you mentioned. Clearly, $A\subset B$. The tricky part here is to show that $B$ is a convex set (needs a little work but surely you can do it). Using the definition of convex hull, it follows that 
$$\operatorname{co}(A)\subset B.$$ 
Next, for any arbitrary convex set $C$ with $A\subset C$, then as what you have said, $B\subset C$. Hence 
$$B\subset\bigcap_{A\subset C}C=\operatorname{co}(A).$$
