Vertical Section preserves Increasing Sequences of Functions

Theorem
Let $X$ and $Y$ be sets.

Let $\sequence {f_n}_{n \mathop \in \N}$ be an increasing sequence of real-valued functions with $f_i : X \times Y \to \overline \R$ for each $i$.

Let $x \in X$.

Then the sequence $\sequence {\paren {f_n}_x}_{n \mathop \in \N}$ is increasing, where $\paren {f_n}_x$ denotes the $x$-vertical section of $f_n$.

Proof
Since $\sequence {f_n}_{n \mathop \in \N}$ is an increasing sequence of real-valued functions, we have:


 * $\map {f_i} {x, y} \le \map {f_j} {x, y}$ for all $i, j$ with $i \le j$.

for all $\tuple {x, y} \in X \times Y$.

In particular, for fixed $x \in X$, we have:


 * $\map {f_i} {x, y} \le \map {f_j} {x, y}$ for all $i, j$ with $i \le j$ and all $y \in Y$.

From the definition of the $x$-vertical section, we have:


 * $\map {\paren {f_i}_x} y \le \map {\paren {f_j}_x} y$ for all $i, j$ with $i \le j$ and all $y \in Y$.

So:


 * $\sequence {\paren {f_n}_x}_{n \mathop \in \N}$ is an increasing sequence of real-valued functions.