Every continuous function bounded implies compact

It occurs to me that it might be nice to post solutions to miscellaneous mathematical exercises at least once in a while.

I saw this one on a chalkboard earlier today; evidently the room was serving as the venue for an analysis class. It’s exactly the sort of elementary exercise that usually takes me a day to solve, if I’m lucky. But this time, I’m happy to report, I managed to figure it out in just a few minutes (while ostensibly listening to a lecture on something else).

Problem: Let be such that every real-valued continuous function on is bounded. Prove that is compact.

25 Potentially Controversial Opinions

I will probably live to regret this, but…

In response to a request that was put to me on a certain social-networking site that will remain nameless…

Here are 25 propositions that I endorse, but which I expect are capable of provoking argument. They are in no particular order.

***
1.There are no deities; all major religions are mistaken.

2.“Atonal” music is no such thing; it is merely highly complex “tonal” music. Schoenberg’s early works are music-theoretically more similar to his middle and later works than to the music of the eighteenth century.

3. Rameau’s theory of “harmony” was wrong from the start; over time it has gotten even wronger.

4. Schenkerian theory, though originally a huge step in the right direction, is now an anachronism; it has been superseded by Westergaardian theory.

5.In exactly the same sense in which there is progress in the sciences, so too is there progress in the arts. The best composers of today know more about music than Beethoven, just as the best mathematicians of today know more about mathematics than Gauss. This doesn’t undermine Beethoven’s greatness any more than it undermines Gauss’s.

6. The role of “natural talent” in the intellectual pursuits is misunderstood and greatly overestimated. You may never learn to paint like Leonardo, but you may indeed come close enough that your previous self couldn’t tell the difference.

7. People who talk about “g factor” with any regularity tend to be jerks.

8. The nature of language has been very poorly understood by philosophers, and only marginally better (on average) by linguists.

9. The error of logical positivism was not the verifiabilty criterion itself, but the fact that it was formulated as a criterion for meaning. What they meant to say was this (and this).

10. The main purpose of studying mathematics is to develop intellectual agility. Mathematicians are (or should be) people who get the same kind of pleasure from manipulating concepts that children do from playing with colored blocks.

11. Intellectuals have a blind spot when it comes to politics. Last year, people who think for a living were capable of arguing that the economy was good under Clinton and bad under Bush, hence you should vote for Obama.

12. Climate science is young and presumably involves some bad-ass partial differential equations; it should not be compared to Darwin’s theory of evolution during policy debates on global warming.

13. Teachers of music theory are perpetually embarrassed by their inability to pinpoint what it is they are trying to teach students. But the test is quite simple: you will know you have done your job when your students can accurately write down music they hear.

14. Most people who wax on about the greatness of Bach’s fugues wouldn’t be able to tell the difference between a fugue by Bach and a fugue by a mediocre contemporary — let alone one by, say, Handel.

15. Richard Dawkins should be knighted.

16. Max Tegmark basically has the right idea with his Level IV Multiverse. (This is my preferred solution to the problem of cosmological fine-tuning.)
.
17. Course grades should be abolished, especially in graduate school. Oral exams/interviews, work samples, and recommendations are entirely sufficient for academic evaluation. (In short: follow the Princeton model.)

18. The “many-worlds” interpretation of quantum mechanics is the correct one.

19. Semantic externalism is so, so wrong. (See here.)

20. There’s really no downside to signing up for cryonics. (Assuming you can overcome enough inertia to actually do so, of course.)

21. General topology is an example of what the definitive solution to a time-honored philosophical problem (the nature of space and continuity) looks like. It is an abstract subject that should blow your mind. It is emphatically not a mere vocabulary that is useful for analysis or algebraic topology.

22. The argument about the Axiom of Choice is over. AC won. It’s one of the Official Axioms of Mathematics. If you don’t like its consequences, see a therapist. (Or: study alternative systems. But for God’s sake don’t claim that standard mathematics is “wrong”.)

23. Finitism is not only misguided, it’s philistine. (OK, so there are only a finite number of atoms in the universe. So the @#$% what? We’re talking about math here.)

24. Karl Popper missed the point. We don’t need to philosophically distinguish “science” from “pseudoscience”; it suffices to distinguish good theories from bad.

25. The phrase “classical music” should be banned. The term is “art music”.

***