Horizontal Section preserves Pointwise Limits of Sequences of Functions

Theorem

Let $X$ and $Y$ be sets.

Let $f : X \times Y \to \overline \R$ be a function.

Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence of functions converging pointwise to $f$.

Let $y \in Y$.

Then:

$\paren {f_n}^y \to f^y$

pointwise, where:

$\paren {f_n}^y$ denotes the $y$-horizontal section of $f_n$

$f^y$ denotes the $y$-horizontal section of $f$.

Proof

From the definition of pointwise convergence, we have:

$\ds \map f {x, y} = \lim_{n \mathop \to \infty} \map {f_n} {x, y}$

for each $x \in X$ and $y \in Y$.

Fix $y \in Y$.

From the definition of the $y$-horizontal section, we have:

$\map {f_n} {x, y} = \map {\paren {f_n}^y} x$

and:

$\map f {x, y} = \map {f^y} x$

So:

$\ds \map {f^y} x = \lim_{n \mathop \to \infty} \map {\paren {f_n}^y} x$

for each $x \in X$.

So:

$\paren {f_n}^y \to f^y$

pointwise.

$\blacksquare$