The question states:
Suppose $x_{1}, x_{2}, ..., x_{k}$ are $k$ points in $\mathbb{R}^n$. Show that the set of all convex combinations of these points,
$$C=\{\sum\limits_{i=1}^{k} c_{i}x_{i} \mid \sum\limits_{i=1}^{i=k} c_{i}=1, c_{i} \geq 0 \text{ for all } i=1,2,...,k \}, $$
is a convex subset of $\mathbb{R}^n$.
Let $y_1,y_2\in C$ and $\alpha \in (0,1).$ Hence there exists $c_1\ldots,c_k, p_1,\ldots,p_k\geq 0$ such that $$\sum_{i=1}^kc_i=\sum_{i=1}^kp_i=1$$ and
$$y_1=\sum_{i=1}^kc_i x_i,\; y_2=\sum_{i=1}^kp_i x_i.$$ Now
$$\alpha y_1+(1-\alpha)y_2= \alpha \sum_{i=1}^kc_i x_i +(1-\alpha) \sum_{i=1}^kp_i x_i$$ $$= \sum_{i=1}^k (\alpha c_i +(1-\alpha)p_i) x_i.$$ Now note that $q_i=\alpha c_i +(1-\alpha)p_i\geq 0$ and sum up to 1.