Bolzano-Weierstrass Theorem and Pareto Efficient Allocation Wikipedia says 'The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation.' 
However, I couldn't find a compelling simple proof for this theorem anywhere. Could someone provide a proof for the theorem in a simple fashion that is understandable for Undergrad students in Economics who have done a course in Analysis?
What one needs are two facts:


Scalarization/Sufficiency for Pareto Optimality
Suppose the economy has finitely many agents $u_i$, $i = 1, \cdots, I$, with feasible allocations given by some $X \subset \mathbb{R}^I$. If
$x = (x_1, \cdots, x_I) \in X$ solves the problem
$$
\max_{x \in X} \sum_{i=1}^I \lambda_i u_i(x_i),
$$
where $\lambda_i &gt; 0$ for all $i$, then $x$ is Pareto optimal. This maximization problem is called the social planner's problem with social weights $\lambda_i$.
(The converse is not true. In other words, not all Pareto allocations arise as such, not even when $u_i$'s are concave.)
Continuous functions map compact sets to compact sets. (The Bolzano–Weierstrass gives a characterization of compactness on $\mathbb{R}^n$---closed and bounded.) 


So if the preferences are continuous and feasible set $X$ is compact, existence of Pareto optimal allocation follows immediately. Choose any social weights $\lambda_i &gt; 0$. A solution of the resulting social planner's problem is a Pareto optimal allocation.