Proof that weak Monotonicity and local non satiation imply monotonic preference

Question

Weak monotonicity in my case is defined as follows: If x is weakly larger than y, then x must be weakly preferred over y. Monotonicity is defined as follows: If x is strictly larger than y, then x must be strictly preferred over y.

So my though process was the following (to proof that weak monoton. + LNS (Local non satiation) implies monotoncity):

LNS tells us that there must exist a y that is strictly preferred over x if this x is in the euclidean distance <ϵ to x. We know that if next to x there is another bundle that is weakly larger than x. Due to weak monotonicity this bundle is weakly preferred over x. Together with LNS monotonicity should hold, which makes intuitively sense but I have no idea how to proof this analytically.

Can somebody help me?

Thank you

Answer 1

The following needs some gaps to be filled in:

Let $x'$ be strictly larger than $x$. We have to show that $x'\succ x$. Let $\epsilon>0$ be small enough that every point of distance less than $\epsilon$ is weakly smaller than $x'$. Let $x''$ be a bundle that has distance less than $\epsilon$ from $x$ but that is strictly preferred to $x$. Such a bundle exists thanks to local nonsatiation. Since $x'$ is weakly larger than $x''$, it is weakly preferred by weak monotonicity. By transitivity, $x'\succ x$.