Suppose $x_1,x_2,\cdots,x_k$ is any finite set of vectors in $\mathbb{R^n}$. A convex combination of these vectors is $\sum_{i=1}^{k}\lambda_ix_i$ where $\sum_{i=1}^{k}\lambda_i=1$ and $\lambda_1,\lambda_2,\cdots,\lambda_k\geq 0$. The set $C$ which contains all the convex combinations is called convex hull.
Show that $C$ is compact in $\mathbb{R^n}$.
I know that since the underlying space is $\mathbb{R^n}$, using Heine-Borel theorem we need to show the boundedness and closedness. For the boundedness if I assume the 2-norm of $x_i$'s is bounded, i.e., $\|x_i\| \leq \|M\|$ then
$$ x=\sum_{i=1}^{k}\lambda_ix_i \leq \sum_{i=1}^{k}\lambda_i \|M\|=\|M\| $$
But for the closedness we need a sequence $(x_k)$ in $C$ and show this converges in $C$. How to show this part?