# Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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### Proving that $O(n)$ is compact

Let $O(n)$ denote the group of orthogonal matrices under multiplication. We want to show that this is set is compact. To show $O(n)$ is compact, we can use Heine-Borel and show that it is closed and ...
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### How do I show $(a+b)^K=\sum _{n=0}^K \binom{K}{n} b^n a^{K-n}$ without using recurrence?

We already know that we can represent this binomial as the following: $$(a+b)^K=\sum _{n=0}^K \binom{K}{n} b^n a^{K-n};$$ where $\binom{K}{n} = \frac{K!}{n! (K-n)!}$ My question here is :How do I ...
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### Show that $\sqrt{6+4\sqrt{2}}-\sqrt{2}$ is rational using the rational zeros theorem [duplicate]

Let $r=\sqrt{6+4\sqrt{2}}-\sqrt{2}$, then $r+\sqrt{2}=\sqrt{6+4\sqrt{2}}$. Squaring both sides, we get $$r^2+2r\sqrt{2}+2=6+4\sqrt{2}$$ which is the same as $r^2-4=2\sqrt{2}$. Squaring both sides ...
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### Is my logic on general Proof-Solving techniques correct?

I've just recently started working through proofs for what's really the first time in my life. Throughout high school, and thus far in college I've never really had to prove things too often, and if I ...
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### Show that $\sqrt{4+2\sqrt{3}}-\sqrt{3}$ is rational using the rational zeros theorem

What I've done so far: Let $$r = \sqrt{4+2\sqrt{3}}-\sqrt{3}.$$ Thus, $$r^2 = 2\sqrt{3}-2\sqrt{3}\sqrt{4+2\sqrt{3}}+7$$ and ...
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### Proving there exist an interval and a number $p$ where $f(x) \leq x^p$ holds

We have a function $f:\mathbb{R}\to [0,1]$, where $f(x)=0$, for $x\leq 0$, and $f$ is a right-continuous function. How can we prove that there exist a number $0 \lt p \leq 1$ and a number $0\lt a$, ...
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### Prove $1^3+2^3+…+n^3 = (1+2+…+n)^2$ for all positive integers $n$. [duplicate]

My approach is to solve this by induction. Base case: $n=1$ $1^3 = 1^2 = 1$ Inductive Step: Suppose that $1^3+2^3+...+n^3 = (1+2+...+n)^2$ holds for all positive integers $n$. We use that to ...
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### When has one sufficiently mastered an area of mathematics?

This is a rather soft question regarding the mastery of various mathematical subjects, such as undergraduate subjects. In particular, say, when has one mastered undergraduate analysis? Is it ...
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### Operator norm and equivalent definitions

From the definition of the operator norm, we have: $||T||_{op}=\inf\{c\in \mathbb{R}^+:||Tv||\leq c||v||, v \in V\}$ If $T: V \rightarrow W$ is a linear map between two normed vector spaces. I have ...
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### Show that $A \lor B ⊢ B \lor A$

Prove the following derivability claim using only our primitive rules: $A \lor B ⊢ B \lor A$ I know this can be attributed to the commutative property, but I'm not exactly sure how to prove this ...
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### Proving that $A \Delta C \subset (A \Delta B) \cup (B \Delta C)$

For two sets we define $A \Delta B = (A \cup B) \setminus (A \cap B)$ (symmetric difference). Problem: Proof that \begin{align*} A \Delta C \subset (A \Delta B) \cup (B \Delta C). \end{align*} ...
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### Proving that $\log_2 7$ is irrational

Prove that $\log_2 7$ is irrational. Book solution: Suppose $\log_2 7$ is rational. Then $\log_2 7=a/b$, where $a$ and $b$ are integers. We may assume that $a>0$ and $b>0$. We have $2^{a/b}=7$, ...