I'm referencing the book, Advanced Calculus, 3rd ed., by Creighton Buck. I feel like the proof is fallacious, but want to confirm my suspicion. I'll state one direction of the proof first, and then call out the particular point where I feel it is fallacious.
Let $f$ be a function defined on a compact set $D$ in $n$ space and taking values in $m$ space, and let $G$ be its graph. Then $f$ is continuous on $D$ if and only if $G$ is compact.
Let the graph of $f$ be
$$G = \{\text{all } P = (p,q) \text{ where } q = f(p), \text{ and } p \in D\}$$
Assume that $G$ is compact. We want to show that $f$ must be continuous. Suppose that this were false. Then there would be some point $p_0$ in $D$, and a sequence $\{p_n\}$ in $D$ with $p_n \rightarrow p_0$, and an $\epsilon > 0$ such that $|f(p_n) – f(p_0)| > \epsilon$ for all $n$. Put $q_n = f(p_n)$ and consider the points $P_n = (p_n, q_n)$ in $G$, for $n = 1, 2, …$. Note that $p_n \rightarrow p_0 \in D$, but that $|q_n – f(p_0)| > \epsilon$. Because $G$ is compact, the sequence $\{P_n\}$ must have a subsequence $\{P_{n_k}\}$ that converges to some point $(p, q)$ in $G$. Accordingly, $\lim_{n \rightarrow \infty}p_{n_k} = p$ and $\lim_{n \rightarrow \infty}q_{n_k} = q$. However, since $\{p_{n_k}\}$ is a subsequence of $\{p_n\}$, which itself converges to $p_0$, we have $p = p_0$. Because the point $(p, q)$ in in $G$, which is the graph of $f$, $q = f(p) = f(p_0)$. However, $\lim_{n \rightarrow \infty}q_{n_k} = f(p_0)$ and $|q_n – f(p_0)| > \epsilon$ are not both possible. We thus conclude that $f$ is continuous on $D$.
I take issue with this statement in the proof:
If $f$ were not continuous, then there would be some point $p_0$ in $D$, and a sequence $\{p_n\}$ in $D$ with $p_n \rightarrow p_0$, and an $\epsilon > 0$ such that $|f(p_n) – f(p_0)| > \epsilon$ for all $n$.
Buck is using the convergence preserving property of continuous functions, which states that a function $f$ is continuous if and only if for any sequence $p_n \rightarrow p$, then $f(p_n) \rightarrow f(p_0)$. Why does this inequality hold true for all $n$?
I feel like this should say:
If $f$ were not continuous, then there would be some point $p_0$ in $D$, and a sequence $\{p_n\}$ in $D$ with $p_n \rightarrow p_0$, and an $\epsilon > 0$ such that for any $N$, $\exists n > N$ such that $|f(p_n) – f(p_0)| > \epsilon$.
With this correction, however, I'm not able to continue with his line of reasoning to arrive at the proof.
Best Answer
Asserting that $f$ is not continuous at $p_0$ means that$$(\exists\varepsilon>0)(\forall\delta>0)(\exists p\in D):\|p-p_0\|<\delta\wedge\bigl\|f(p)-f(p_0)\bigr\|\geqslant\varepsilon.$$So, take such a $\varepsilon>0$ and take $\varepsilon'\in(0,\varepsilon)$. Let $n$ be a natural number. Then there's a $p_n\in D$ such that $\|p_n-p_0\|<\frac1n$ and $\bigl\|f(p_n)-f(p_0)\bigr\|\geqslant\varepsilon>\varepsilon'$. So, $\lim_{n\to\infty}p_n=p_0$ and$$(\forall n\in\mathbb{N}):\bigl\|f(p_n)-f(p_0)\bigr\|>\varepsilon'.$$