<p>By local nonsatiation there exists a sequence <span class="math-container">$x_n\rightarrow x$</span> with <span class="math-container">$x_n\succ x\,\,\forall n$</span>. By transitivity <span class="math-container">$x_n\succ x^*\,\,\forall n$</span>. By contraposition <span class="math-container">$p\cdot x_n&gt;m\,\,\forall n$</span>. By continuity of the dot product <span class="math-container">$p\cdot x\ge m$</span>.</p>
<p>This is pretty much a somewhat more compact version of your own approach.</p>

CC BY-SA 4.0
2021-03-01T22:58:33.223Z
2021-03-01T22:58:33.223Z
VARulle
<p>Suppose that <span class="math-container">$x^*$</span> satisfies <span class="math-container">$x^*\succsim x$</span> for <span class="math-container">$\forall x\in\{{x∈X|p·x\leq m}\}$</span>.</p>
<p>How can we prove that <span class="math-container">$x\succsim x^*$</span> <span class="math-container">$\Rightarrow$</span> <span class="math-container">$p·x≥m$</span> if <span class="math-container">$\succsim$</span> is locally nonsatiated?</p>
<p>My idea for this goes like this but I am not sure:</p>
<p>Suppose that there exists <span class="math-container">$x' ∈ X$</span> such that <span class="math-container">$x'\succsim x^∗$</span> and <span class="math-container">$p · x' &lt; m.$</span> Since we have local nonsatiation, for any <span class="math-container">$\epsilon&gt;0$</span> there exists <span class="math-container">$x''\in X$</span> such that <span class="math-container">$||x''−x'||≤\epsilon$</span> and <span class="math-container">$x''\succ x'$</span>. By transitivity of <span class="math-container">$\succsim$</span> and by <span class="math-container">$x''\succ x'\succsim x^*$</span>. The former implies, by <span class="math-container">$p·x' &lt; m$</span>, that we can choose <span class="math-container">$\epsilon$</span> small enough such that <span class="math-container">$p·x'' &lt; m$</span> and hence <span class="math-container">$x''\in\{{x∈X|p·x\leq m}\}$</span>. This implies, by the definition of <span class="math-container">$x^∗$</span>, that <span class="math-container">$x^∗\succsim x''$</span>, a contradiction.</p>

CC BY-SA 4.0
2021-03-01T19:31:21.523Z
Your idea is largely correct, but there are a few typos. (1) In the 2nd sentence of the last paragraph, $x$ should have been $x'$; (2) where transitivity is invoked, the chain should have been $x''\succ x'\succsim x^*$; and (3) there's an extraneous "$\forall x \in$" in the second last sentence specifying the set to which $x''$ belongs.
Herr K.
CC BY-SA 4.0
2021-03-01T19:42:46.327Z
I fixed them. I hope that it is correct now. Thank you for your input!
Maybeline Lee
<microeconomics><preferences>
Local nonsatiation
Maybeline Lee
# Local nonsatiation

(Asked by: username_0 on 2021-03-01)

Suppose that $x^\*$ satisfies $x^\*\succsim x$ for $\forall x\in\{{x∈X|p·x\leq m}\}$.

How can we prove that $x\succsim x^\*$ $\Rightarrow$ $p·x≥m$ if $\succsim$ is locally nonsatiated?

My idea for this goes like this but I am not sure:

Suppose that there exists $x' ∈ X$ such that $x'\succsim x^∗$ and $p · x' < m.$ Since we have local nonsatiation, for any $\epsilon>0$ there exists $x''\in X$ such that $||x''−x'||≤\epsilon$ and $x''\succ x'$. By transitivity of $\succsim$ and by $x''\succ x'\succsim x^\*$. The former implies, by $p·x' < m$, that we can choose $\epsilon$ small enough such that $p·x'' < m$ and hence $x''\in\{{x∈X|p·x\leq m}\}$. This implies, by the definition of $x^∗$, that $x^∗\succsim x''$, a contradiction.

**Tags:** microeconomics, preferences
> **Comments:**
> **username_1**: Your idea is largely correct, but there are a few typos. (1) In the 2nd sentence of the last paragraph, $x$ should have been $x'$; (2) where transitivity is invoked, the chain should have been $x''\succ x'\succsim x^*$; and (3) there's an extraneous "$\forall x \in$" in the second last sentence specifying the set to which $x''$ belongs.  
> **username_0**: I fixed them. I hope that it is correct now. Thank you for your input!  

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## 1 answers

### Answer by username_2 on 2021-03-01. Score: 5 | Accepted answer

By local nonsatiation there exists a sequence $x\_n\rightarrow x$ with $x\_n\succ x\,\,\forall n$. By transitivity $x\_n\succ x^\*\,\,\forall n$. By contraposition $p\cdot x\_n>m\,\,\forall n$. By continuity of the dot product $p\cdot x\ge m$.

This is pretty much a somewhat more compact version of your own approach.


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