# Tagged Questions

For questions about the formulation of a proof, not about the mathematics behind it.

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### Proving by induction that a balanced strings of parentheses has equally many opening and closing parentheses

In computer science, a string is a finite sequence of characters. For strings $A$ and $B$, we express $AB$ as $A$ followed by $B$. A balanced string of parentheses is a string of open and closed ...
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### Prove the following simple exponentiation equality.

Having trouble with the following proof. Given $b > 1, c > 0$, prove that $\exists \; x$ s.t. $b^{x} < c$. We can't use $log$, and I have already shown that $b^{x} > c$ by using the ...
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### Theorem 2.27 (a) in Baby Rudin: Is his proof complete enough?

Here's Theorem 2.27 (a) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $X$ is a metric space and $E \subset X$, then $\overline{E}$ is closed. Now here's ...
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### Prove $\log(x)^{10} < x$ (for $x > 10^{10}$)

I need to prove that $\log(x)^{10} < x$ for $\ x>10^{10}$ It's pretty clearly true to me, but I need a good proof of it. I tried induction, and got stuck there.
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### Understanding a proof about nested nonempty connected compact subsets

I know this question has asked to death here on MSE but I have not found a satisfactory solution. A solution found online is extremely elegant but I do not quite understand it! Given nested ...
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### Proving two integers of opposite parity have an even product?

I think I might be beginning to wrap my head around some simpler proofs, but I'm a little stumped on this one from my textbook: Use a direct proof to show that if two integers have opposite ...
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### For all sets A and B, if A ⊆ B, then A ∪ B ⊆ B

I am trying to solve a proof, but I'm a little lost on how to structure it. I have the following setup, but I'm not sure what to put in most of the blank spaces. Proposition: For all sets A and B, ...
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### prove that $-1 \le \frac{ac+bd}{\sqrt{a^2+b^2}\sqrt{c^2+d^2}} \le 1$

For the real numbers $a, b, c, d$ prove that $$-1 \le \dfrac{ac+bd}{\sqrt{a^2+b^2}\sqrt{c^2+d^2}} \le 1$$ Actually if we let $\vec{u} = (a, b)$ and $\vec{v} = (c, d)$ then by dot product we got ...
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### Sufficient condition for upper semicontinuous functions

My question might be fundamental but I'm glad if you give some help since I don't find any idea. Let $X$ be a bounded and closed subset of $\mathbb{R}$. A function $f:X\to\mathbb{R}$ is called to be ...
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### Proving the Trichotomy Property

I need to show that if $a,b\in \mathbb{R}$, then only one of the following holds: $a\in \mathbb{P}, -a\in \mathbb{P}$, or $a=0.$ By a definition in my book, if $a-b \in \mathbb{P}$, then $a>b$; ...
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### Upper estimate between an original function and its sup-convolution under a limitation

My setting maybe look rather special but I'm glad if you give some answers. Let $f:[0,1]\to\mathbb{R}$ be a bounded, upper semicontinuous function and $f^{\varepsilon}:[0,1]\to\mathbb{R}$ be $f$'s ...
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### Is it correct? $1^n +2^n +…+(p-1)^n=-1 \pmod p$

$p$ a prime number, $n\in \mathbb{N}$ and $p-1\mid n$ then $1^n +2^n +...+(p-1)^n=-1 \pmod p$ I'm not sure if my proof is correct: Take the group $G=(\mathbb{Z*}_{p},\cdot)$ with the ...
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### Simple Proof for Commutative Property of Multiplication

I'm supposed to show that $a\cdot b=b\cdot a$ for a set $K:=\{s+t\sqrt2:s,t\in\mathbb{Q}\}$ to show that this set is a field. I was going to set it up like: Let $a, b\in K$ such that ...
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### How to prove that bijective functions have surjective inverses

I am trying to prove Since $f$ is bijective therefore it has an inverse and $f^{-1}$ is bijective But stuck on proving that $f^{-1}$ is surjective Suppose $f: A \to B$ is a bijection, then a ...
Let G be a simple graph with degree sequence $(d_1,d_2,...,d_n)$. Prove that for each k, $0<k<n$: $$\sum_{i=1}^k d_i\le k(k-1)+\sum_{i=k+1}^n min(k,d_i)$$ I'm new to graph theory and proof ...
### Can I use “$\iff$” symbol when I “transform” an expression to another form?
I am writing a solution to prove that $\sqrt5$ is not rational. Here is my first half proof: Assume $\sqrt{5}$ is a rational number. By the definition of rational number, $\sqrt{5} = \frac{p}{q}$, ...