This short post aims at proving a general version of the Moore-Osgood Theorem on interchanging of limits.
Let $X$ be a complete metric space; we will use $|\bullet -\bullet|$ for the distance function on $X$. Let $\Delta, \Gamma$ be two directed sets. Given a functions $f:\Delta \times \Gamma \to X$, $g: \Delta \to X$, and $h:\Gamma \to X$. Suppose
- $\lim_{\Delta} f = h$ uniformly, and
- $\lim_{\Gamma} f = g$ pointwise.
Then $\lim_\Gamma h = \lim_\Delta g = \lim_{\Delta \times \Gamma} f$, where $\Delta\times \Gamma$ is the equipped with the Cartesian product direct set structure $(\delta_1, \gamma_1) \leq (\delta_2, \gamma_2) \iff \delta_1 \leq \delta_2 \wedge \gamma_1 \leq \gamma_2$.
(The theorem also works for uniform spaces; the proof just requires making the obvious changes.)
Our assumptions imply:
- There exists a function $\delta_{fh}(\epsilon)$ such that for every $\delta \geq \delta_{fh}(\epsilon)$, and for every $\gamma\in \Gamma$, we have $|f(\delta, \gamma) - h(\gamma)| < \epsilon$.
- There exists a function $\gamma_{fg}(\epsilon,\delta)$ such that for every $\gamma \geq \gamma_{fg}(\epsilon,\delta)$, we have $|f(\delta, \gamma) - g(\delta)| < \epsilon$.
First, we prove that $h$ is Cauchy. Let $\epsilon > 0$. Consider \[ \gamma_1, \gamma_2 \geq \gamma_{fg}(\epsilon/4, \delta_{fh}(\epsilon/4)). \] Then \[ |h(\gamma_1) - h(\gamma_2)| \leq |h(\gamma_1) - f(\delta_{fh},\gamma_1)| + |f(\delta_{fh}, \gamma_1) - g(\delta_{fh})| + |g(\delta_{fh}) - f(\delta_{fh}, \gamma_2)| + |f(\delta_{fh},\gamma_2) - h(\gamma_2)| < 4 \times \frac{\epsilon}{4} = \epsilon. \] This shows that $h$ is Cauchy. By completeness of $X$ the limit $\lim_\Gamma h = L$ exists. That is to say, we further have
- There exists a function $\gamma_h(\epsilon)$ such that for all $\gamma \geq \gamma_h(\epsilon)$, the distance $|L - h(\gamma)| < \epsilon$.
Next we show that $g$ and $f$ both converge also to $L$. Let $\epsilon > 0$, choose \[ \delta_1 \geq \delta_{fh}(\epsilon/3) \text{ and } \gamma_1 \geq \max( \gamma_{fg}(\epsilon/3,\delta_1), \gamma_h(\epsilon/3)).\] Observe that \[ |g(\delta_1) - L| \leq |g(\delta_1) - f(\delta_1, \gamma_1)| + |f(\delta_1, \gamma_1) - L| < \frac{\epsilon}{3} + |f(\delta_1, \gamma_1) - L| \] and \[ |f(\delta_1, \gamma_1) - L| \leq |f(\delta_1, \gamma_1) - h(\gamma_1)| + |h(\gamma_1) - L| < \frac{2\epsilon}{3}.\] This shows the desired convergence.
