# Equivalence of Definitions of Continuity of Preferences

We have two definitions of the continuity of preferences:

Def 1: $$\succcurlyeq$$ is continuous if for any sequences $$\{x^n\} \subset X$$ and $$\{y^n\} \subset X$$, then $$n \in \mathbb{N}$$ such that,

• $$\forall n, \quad x^n \succcurlyeq y^n$$
• $$\lim_{n \rightarrow \infty} x^n = x, \quad \lim_{n \rightarrow \infty} y^n = y \quad (\text{where x, y \in X)}$$

then $$x \succcurlyeq y$$

and

Def 2: $$\succcurlyeq$$ is continuous if whenever $$x \succ y$$, $$\exists \ B_x, \ B_y$$, open balls around $$x, y$$, such that $$\forall \ x' \in B_x, \ y' \in B_y$$, then we have $$x' \succ y'$$.

Show that the following definitions are equivalent to each other.

We want to show that for $\succcurlyeq$ on $X$, Def 1 $\iff$ Def 2

$\boxed \Longrightarrow$

Assume that $\succcurlyeq$ is continuous by Def 1.

Let us say $x \succ y$. Denote our open-balls as $B(x, r)$, an open ball around $x$ of radius $r$. Suppose $\forall n, \ \exists \ x^n \in B(x, \frac{1}{n}), \ y^n \in B(y, \frac{1}{n})$ such that $y^n \succcurlyeq x^n$. But then we have constructed $\{x^n\} \rightarrow x$ and $\{y^n\} \rightarrow y$, and by Def 1, $y \succcurlyeq x$, which is a contradiction.

$\boxed \Longleftarrow$

Assume that $\succcurlyeq$ is continuous by Def 2.

Let us take sequences ${x^n} \subset X$ and ${y^n} \subset X$ where $\forall n, \quad x^n \succcurlyeq y^n$ and $\lim_{n \rightarrow \infty} x^n = x, \quad \lim_{n \rightarrow \infty} y^n = y \quad (\text{where$x, y \in X)$}$ for $n \in \mathbb{N}$,

BUT $x \succcurlyeq y$ is false instead of true. We want to show this leads to a contradiction.

If $x \succcurlyeq y$ is false, (so $y \succ x$) then $\exists B_x, B_y$ such that $\forall y' \in B_y, x' \in B_x$, we have $y' \succ x'$. Because $\{x^n\} \rightarrow x, \{y^n\} \rightarrow y$, there exists $N$ large enough such that $\forall n>N$, we have $y^n \in B_y, x^n \in B_x$.

Thus $\forall n > N$, we have $y^n \succ x^n$, which contradicts $\forall n, \ x^n \succcurlyeq y^n$.