Convex Polytope closedness

[zcd]

Homework Statement

Prove that every convex polytope is convex and closed.

Homework Equations

$$C=\{ \sum_{j=1}^n x_j a^j | x_j \geq 0, \sum_{j=1}^n x_j = 1\}$$ is a convex polytope

The Attempt at a Solution

I've already proven the convexity portion. To prove C is closed, I let $\{ b^N \}_{N=1}^\infty \subseteq C$ and assumed $\lim_{N\to\infty} b^N = b$.
$b=\sum_{j=1}^n x_j a^j$, so I have to show $\lim_{N\to\infty} x^N = x$.

I started with $x_j \geq 0, \sum_{j=1}^n x_j = 1\$ means $|x^N| \leq 1$ and the sequence $\{ x^N \}_{N=1}^\infty$ is a bounded sequence. From here, I can use the Bolzano-Weierstrass theorem to show that there exists a subsequence that converges. From here, I'm unsure of what to do because the subsequence converges to some value which may or may not be the right value

 

[Office_Shredder]

the subsequence converges to some value which may or may not be the right value

You know that the whole sequence converges to b, so every subsequence converges to b as well.