Simplify my proof of the continuity of entropy

_psets/3.md

@@ -38,50 +38,12 @@ function on its domain. Then $$x \log(x)$$ is also continuous, since finite prod
 functions are continuous. This suffices for $$x > 0$$. At zero, $$x \log x$$ is continuous because
 we have defined it to be equal to the limit we found above.
 
-Thus each term of the entropy function is a continuous function from $$\mathbb{R}$$ to
-$$\mathbb{R}$$. This suffices to show that negative entropy is continuous, based on the lemma below.
-Thus entropy is continuous, since negation is a continuous function, and finite compositions of
-continuous functions are continuous.
-
-The necessary lemma is easy to prove, if symbol heavy. I will use the
-[$$L^1$$ norm](https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm), but this is without loss of
-generality because all norms on a finite dimensional vector space induce the same topology. Let
-$$f : \mathbb{R}^n \to \mathbb{R}$$ and $$g : \mathbb{R} \to \mathbb{R}$$ be continuous functions,
-and define $$h : \mathbb{R}^{n + 1} \to \mathbb{R}$$ as
-
-$$h(x_1, \ldots, x_{n + 1}) = f(x_1, \ldots, x_n) + g(x_{n + 1})$$
-
-Fix any $$x = (x_1, \ldots, x_{n + 1}) \in \mathbb{R}^{n + 1}$$, and any $$\epsilon > 0$$. Since
-$$f$$ is continuous, there exists some positive $$\delta_f$$ such that for any $$y \in
-\mathbb{R}^n$$, $$\lVert (x_1, \ldots, x_n) - (y_1, \ldots, y_n) \rVert < \delta_f$$ implies
-$$\lVert f(x_1, \ldots, x_n) - f(y_1, \ldots, y_n) \rVert < \epsilon / 2$$. For the same reason
-there is a similar $$\delta_g$$ for $$g$$. Let $$\delta$$ be the smaller of $$\delta_f$$ and
-$$\delta_g$$. Now fix any $$y \in \mathbb{R}^{n + 1}$$ such that $$\lVert x - y \rVert < \delta$$.
-Note that
-
-$$
-\begin{align*}
-\lVert (x_1, \ldots, x_n) - (y_1, \ldots, y_n) \rVert &= \sum_{i = 1}^n \lVert x_i - y_i \rVert \\
-&\leq \sum_{i = 1}^{n + 1} \lVert x_i - y_i \rVert \\
-&< \delta_f
-\end{align*}
-$$
-
-and similarly for the projections of $$x$$ and $$y$$ along the $$n + 1$$st dimension. Thus
-
-$$
-\begin{align*}
-\lVert h(x) - h(y) \rVert
-&= \lVert f(x_1, \ldots, x_n) + g(x_{n + 1}) - f(y_1, \ldots, y_n) + g(y_{n + 1}) \rVert \\
-&\leq \lVert f(x_1, \ldots, x_n) - f(y_1, \ldots, y_n) \rVert +
-      \lVert g(x_{n + 1}) + g(y_{n + 1}) \rVert \\
-&< \frac{\epsilon}{2} + \frac{\epsilon}{2} \\
-&= \epsilon
-\end{align*}
-$$
-
-It follows that $$h$$ is continuous.
-
+Thus each term of the entropy function is a continuous function from $$\mathbb{R}_{\geq 0}$$ to
+$$\mathbb{R}$$. But we can also view each term as a function from $$\mathbb{R}_{\geq 0}^n$$ to
+$$\mathbb{R}$$. Each one ignores most of its inputs, but this doesn't change its continuity. (The
+epsilon-delta proof is simple, since the only part of the distance between inputs that matters is
+that along the active coordinate.) So entropy is a sum of continuous functions, and is thus
+continuous.
 
 ### Non-negativity