Can somebody tell me why part (2) of this proof is of any relevance?
(1) Assume that $x$ is Pareto-dominated by $y$. Since $x_i$ is optimal for consumer $i$: if $y_i\succ x_i$ then $p\cdot y_i > p\cdot x_i$
(2) Local non-satiation even implies
if $y_i\succeq x_i$ then $p\cdot y_i\geq p\cdot x_i$
(3) Adding up: $p\cdot\sum y_i > p\cdot\sum x_i= p\cdot w$, in violation of feasibility. ($w$= initial endowment)
The most common version of Pareto dominance says that an allocation is Pareto-dominated if there is another feasible allocation in which at least one agent is better off and everyone else is at least as well off. In particular, the latter allocation can include consumers that are just as well off as before.
It follows from the very definition of a competitive equilibrium that (1) holds. That a consumer chooses the best bundle they can afford means that every better bundle must be unaffordable and, therefore, more expensive.
Now, the sum in (3) is not just over those agents that are strictly better off, it is also over those who are equally well off. For those, one only gets the weak inequality (2) using local-nonsation.
Here is an example where local-nonsation is violated and in which a competitive equilibrium is not Pareto-efficient, showing the importance of the assumption of local non-satiation for the first welfare theorem:
There are two goods and two consumers. Both consumers have the initial endowment $\omega_1=\omega_2=(1,1)$. For consumer 1, both goods are perfect substitutes; consumer 1 wants the sum of the amounts of both commodities they consume as large as possible. Consumer 2, does not care what they get, they are indifferent among all consumption bundles. You can check that there is a competitive equilibrium in which both consumers simply receive their endowment and both goods have the same price, $p_1=p_2>0$. But the unique Pareto-efficient allocation gives everything to consumer 1, they receive the bundle $(2,2)$ and consumer 2 the bundle $(0,0)$. Every allocation that Pareto-dominates the proposed equilibrium allocation has consumer 1 being better off and therefore receiving a more expensive bundle (for the equilibrium prices), but consumer 2 receiving a cheaper bundle that is still as good as before.
There is also a weak notion of Pareto-efficiency: An allocation is weakly Pareto-efficient if there is no feasible allocation in which everyone is strictly better off. One can prove that every competitive equilibrium is weakly Preto-efficient even without local non-satiation: If everyone is better off, (1) implies that everyone receives a more expensive bundle and then we can directly go to (3).
Local non-satiation means that you always want a little bit more. There are no sweet spots where you are content and wouldn't accept any more of $x_i$.
What this means in practice is that if you are optimising your allocation of goods $x_i$, you will exhaust all of your resources ($w$).
If you don't have (2), it's possible that your optimal bundle will involve some combination of $x_i$ and $w$.
With (2), your optimal bundle will only contain $x_i$.
