<p>Without any more assumptions, this need not be true. Here is an economically nonsensical but mathematically fine example: Consider the following relation on the real line: <span class="math-container">$x\succeq y$</span> holds if and only if <span class="math-container">$x=y$</span> and <span class="math-container">$x$</span> is a rational number. The graph of this relation is <span class="math-container">$G=\{(q,q)\mid q\in\mathbb{Q}\}$</span>, a set that is clearly not closed. So there exist sequences <span class="math-container">$\langle (x_m,y_n)$</span> converging to <span class="math-container">$(x,y)$</span> satisfying <span class="math-container">$x_n\succeq y_n$</span> for all <span class="math-container">$n$</span> but <span class="math-container">$x\succeq y$</span> does not hold. However, all upper and lower contour sets are empty or contain a single point and are, therefore, closed.</p>

CC BY-SA 4.0
2024-04-29T22:04:57.893Z
2024-04-29T22:50:33.033Z
2024-04-29T22:50:33.033Z
Michael Greinecker
<p>I'm concerned with the reverse direction, that upper and lower contour is closed implies the preference is continuous, that is  for any sequence <span class="math-container">$x_n$</span> and <span class="math-container">$y_n$</span>, <span class="math-container">$x_n\succcurlyeq y_n$</span> for all <span class="math-container">$n$</span>, then if <span class="math-container">$x_n\to x$</span> and <span class="math-container">$y_n\to y$</span>, we have <span class="math-container">$x\succcurlyeq y$</span>.</p>
<p>I have read some of the proofs of it, but it seems all of them have utilised that the preference is transitive. Included the solution manual for MWG, it states use Proposition 1.B.1, which is the proposition under rationality. But isn't it that we don't have rationality here?</p>

<microeconomics><consumer-theory><preferences>
Prove a preference preserved under limits if and only if its upper and lower contour is closed
Nonenicht
# Prove a preference preserved under limits if and only if its upper and lower contour is closed

(Asked by: username_0 on 2024-04-29)

I'm concerned with the reverse direction, that upper and lower contour is closed implies the preference is continuous, that is for any sequence $x\_n$ and $y\_n$, $x\_n\succcurlyeq y\_n$ for all $n$, then if $x\_n\to x$ and $y\_n\to y$, we have $x\succcurlyeq y$.

I have read some of the proofs of it, but it seems all of them have utilised that the preference is transitive. Included the solution manual for MWG, it states use Proposition 1.B.1, which is the proposition under rationality. But isn't it that we don't have rationality here?

**Tags:** microeconomics, consumer-theory, preferences

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## 1 answers

### Answer by username_1 on 2024-04-29. Score: 4 | Accepted answer

Without any more assumptions, this need not be true. Here is an economically nonsensical but mathematically fine example: Consider the following relation on the real line: $x\succeq y$ holds if and only if $x=y$ and $x$ is a rational number. The graph of this relation is $G=\{(q,q)\mid q\in\mathbb{Q}\}$, a set that is clearly not closed. So there exist sequences $\langle (x\_m,y\_n)$ converging to $(x,y)$ satisfying $x\_n\succeq y\_n$ for all $n$ but $x\succeq y$ does not hold. However, all upper and lower contour sets are empty or contain a single point and are, therefore, closed.


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