陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「245A: Problem solving strategies」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

This is going to be a somewhat experimental post. In class, I mentioned that when solving the type of homework problems encountered in a graduate real analysis course, there are really only about a dozen or so basic tricks and techniques that are used over and over again. But I had not thought to actually try to make these tricks explicit, so I am going to try to compile here a list of some of these techniques here. But this list is going to be far from exhaustive; perhaps if

(See also the Tricki for some general mathematical problem solving tips. Once this page matures somewhat, I might migrate it to the Tricki.)

已知结果和反例

Note: the tricks occur here in no particular order, reflecting the stream-of-consciousness way in which they were arrived at. Indeed, this list will be extended on occasion whenever I find another trick that can be added to this list.

If one has to show that two numerical quantities X and Y are equal, try proving that and separately. (Often one of these will be very easy, and the other one harder; but the easy direction may still provide some clue as to what needs to be done to establish the other direction.)

证明或构造的主线

In a similar spirit, to show that two sets E and F are equal, try proving that and .

If one has to show that , try proving that for any . (This trick combines well with Trick 1.)

阅读时建议盯住的点

In a similar spirit, if one needs to show that a quantity vanishes, try showing that for every .

Or: if one wishes to show that two functions agree almost everywhere, try showing first that holds for almost every x, or even just outside of a set of measure at most , for any given .

值得单独记下的条目

• if one has to prove something about a measurable set, try proving it for open, closed, compact, bounded, or elementary sets first.
• if one has to prove something about a measurable function, try proving it for functions that are continuous, bounded, compactly supported, simple, absolutely integrable, etc.
• if one has to prove something about a complex-valued function, try it for real-valued functions first.
• If one has to prove something about a real-valued function, try it for unsigned functions first.
• If one has to prove something about a simple function, try it for indicator functions first.
• A sequence of functions that converges in norm or in measure can be refined to a subsequence that converges pointwise almost everywhere as well.
• A sequence in a (sequentially) compact space may not converge at all, but some subsequence of it will always converge.
• The pigeonhole principle: A sequence which takes only finitely many values has a subsequence that is constant. More generally, a sequence which lives in the union of finitely many sets has a subsequence that lives in just one of these sets.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源