### Point-set topology, a short review

Here, we work through (in review) selected exercises from Chapter 2 of Charles Pugh's Real Mathematical Analysis, in review for a second course on real analysis I am planning to take this coming (junior) fall. At the end, we have a discussion of the Riemann integral and its limitations, in anticipation of the Lebesgue integral, which is introduced in the aforementioned course.

(7) Write the limit of the sequence as $$p$$. Fix some arbitrary $$\epsilon > 0$$ and find $$k \in \mathbb{N}$$ such that $$|p_j - p| < \epsilon \ \forall j > k$$ and split $$(p_n)$$ into two sub-sequences $$(p_0, \cdots, p_k)$$ and $$(p_{k+1},p_{k+2}, \cdots)$$. Then see that we know all points in the second sequence are within $$\epsilon$$ of $$p$$ by construction since $$(p_n)$$ converges to $$p$$. Then see that since the first sequence is finite, we can find a maximum distance of any of those points from $$p$$, and call this distance $$d \in \mathbb{R}$$. Then we can let $$\delta = \text{max}(d, \epsilon)+1$$ to get that $$p_i \in B_\delta(p)$$ for all $$p_i \in (p_n)$$, as required.

(10) Consider, WLOG (the same idea goes for decreasing sequences) an increasing sequence, $$(p_n) \in \mathbb{R}$$. Assume it is bounded. Then by the LUB property of the real numbers, it has a supremum, call it $$p$$. To see this is in fact the limit of the sequence, fix an $$\epsilon > 0$$ and see that eventually $$(p_n)$$ must be $$\epsilon-\text{close}$$ to the LUB because if it was never that close, you could find a tighter upper bound, $$p'$$, for the sequence. This would contradict the fact that $$p$$ is the least upper bound. Therefore $$(p_n)$$ converges to $$p$$, as required.

(11)(a) Let $$(x_n) \in \mathbb{R}$$ be a sequence, and we seek to find a monotone subsequence. Call a natural number $$x_n$$ "nice" if $$m > n \implies x_m > x_n$$. That is, if every number after $$x_n$$ is strictly larger than it. Then see that either our sequence $$(x_n)$$ has finitely many "nice" numbers or not.

If it does, take some $$x_k, k > n$$ for $$x_n$$ the final "nice" number. Then we are guaranteed that there exists some $$x_{k+1}$$ that is smaller than or equal to $$x_k$$ because $$x_k$$ is not nice. And same with $$x_{k+1}$$ inductively, to generate an infinite sequence of non-increasing numbers.

If $$(x_n)$$ has infinitely many "nice" numbers, then those form a non-decreasing sequence since each is greater than the last, by definition of "nice." Therefore, either way, our sequence $$(x_n)$$ has a monotone subsequence, as required.

(b) Every subsequence of a bounded sequence is also bounded, and so if we take a subsequence of $$(x_n)$$ that is monotone, we can use (10) to see that subsequence converges, and thus $$(x_n)$$ has a convergent subsequence.

(c) The fact that every bounded sequence has a convergent subsequence that we just proved above is exactly equivalent to Bolzano-Weierstrass for $$n=1$$.

(d) Heine-Borel states that sequentially compact $$\iff$$ closed and bounded. A metric space $$(X, d)$$ is sequentially compact if every sequence within it converges to a limit in the same metric space. BW just gives that every bounded sequence in the reals has a convergent subsequence, which isn't enough to establish sequential compactness of the reals due to the boundedness condition (in fact, the reals do not comprise a compact set), so this does not help us get a proof of Heine-Borel.

(13) Fix arbitrary $$\epsilon > 0$$ and $$x_0 \in M$$. We seek a $$\delta$$ such that $$|x-x_0| < \delta \implies |f| < \epsilon$$

(19)

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(40)

[TODO. The Riemann Integral and its limitations]

[TODO. Theorems to recap: Rolle, Darboux, IVT, EVT, MVT, Bolzano-Weierstrass, Heine-Borel, Arzela-Ascoli, Inverse Function Theorem, Implicit Function Theorem]