# 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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### Another form of the Sandwich theorem (for derivatives in dimension $1$)

Here is the theorem : "Let $I\subseteq \mathbb{R}$ an interval which contains $a\in \mathbb{R}$. Let $M$ and $m$ two functions defined on $I$, differentiable at $a$ and $f$ a function defined on $I$ ...
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### base change of exterior powers

Let $n\geq 0$ be an integer, $R\to R'$ a ring homomorphism, and $M$ an $R$-module. Then the following holds: $$\bigl(\bigwedge^n_R M\bigr)\otimes_r R' \cong \bigwedge_{R'}^n\, (M\otimes_r R').$$ I ...
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### Proof of the Curtis-Hedlund Theorem: Why is there a function $\mu\colon A^S\to A$ such that $\tau(x)(1_G)=\mu(x_{|S})$ for all $x\in A^G$?

Here is the Curtis-Hedlund Theorem and its proof [the sets $V(\cdot,\cdot)$ used in this proof are explained below.]: My problem is I am not sure that I have understand that correctly. So I ...
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### Question on the Squeeze theorem

In this theorem we consider the functions $f$, $g$ and $h$ which are defined on $\bar {\mathbb{R}}$ except possibly at $a \in \bar {\mathbb{R}}$ but could we have the limit in $a$ of these three ...
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### Inequality.such as Nesbitt

Let $a,b,c >0$ , prove that: $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b} \leq \dfrac{a^2}{b^2+c^2}+\dfrac{b^2}{c^2+a^2}+\dfrac{c^2}{a^2+b^2}$$
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### Shuffling cards and laying them out in order

The numbers from 1 to 50 are printed on cards. The cards are shuffled and then laid out face up in 5 rows of 10 cards each. The cards in each row are rearranged to make them increase from left ...
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### Discrete Mathematics: $mn + 2m + 2n + 2 = n$ proof of uniqueness of $m$, $\forall n \in \mathbb{Z}$

Prove: There exists a unique integer $m$ such that for every integer $n$: $$mn + 2m + 2n + 2 = n$$ However I am not sure if my proof is correct. How do I prove uniqueness of $m$? I prove it by ...
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### Tips on constructing a proof by induction.

So right now I'm working on a discrete mathematics course and I've been having a bit of trouble figuring out how to prove certain equations using mathematical induction. I have very little trouble ...
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### proof that the expression is Real for any $z$

Please help me with this problem, I'm clueless here. $\ \ \ \ (\bar{z}+1-2i)^{1985} + (\bar{z}+1+2i)^{1985}$ $\ \ \ \$proof that the expression is Real for any $z$
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### Question about loss of generality in proofs

My concern is with choosing specific conditions within a proof to arrive at a general result. As an example, I'll use the proof that $\mathbb{Q}$ is dense in $\mathbb{R}$. The proof I know goes as ...
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### methods of proof, discrete mathematics

"Disprove: For all integers $r, m,$ and $n$, if $r$ divides $mn$ then either $r$ divides $m$ or $r$ divides $n$." I am not sure if I am on the right track To disprove I try the negation of a ...
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### Prove that a square of a positive integer cannot end with $4$ same digits different from $0$

Prove that a square of a positive integer cannot end with $4$ same digits different from $0$. I already proved that square of positive integer cannot end with none of digits $1,2,3,5,6,7,8,9$ using ...
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### Solve the equation $2x^2+5y^2+6xy-2x-4y+1=0$

The problem does not say it but I think solutions should be from $\mathbb{R}$. I tried to express the left sum as a sum of squares but that does not work out. Any suggestions?
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### A quadratic polynomial is nonnegative for all $x$ if and only if the discriminant is nonpositive

Show that if $a>0$ the inequality $ax^2+2bx+c\ge 0$ for all values of $x$ if and only if $b^2-ac\le 0$. I tried to prove it by: $ax^2+2bx+c≥ b^2-ac$. Used partial derivatives with respect to ...
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### Conditions so that Lebesgue Covering Dimension and “Usual” Dimension are Equal

The definition of covering dimension is as follows: The ply of a cover is the smallest number $n$ (if it exists) such that each point of the space belongs to at most n sets in the cover. A refinement ...
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### Prove that $V_i$ are $T$-invariant for $1\le i\le k$ and $V=\bigoplus_{i=1}^{k}V_i$

Let $T$ be a linear operator over $V$ with dim$(V)=n$ and let the ordered set $B=${$v_1,v_2,...v_n$} be a basis for $V$. Furthermore, let $A=[T]_B$ be block-diagonal matrix (that is ...
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### Deduce an inequality by using Bernoulli's Inequality

Deduce $c^n\geq c \forall n\mathbb\in{N},c>1$ What I have tried is $$\text{Let }x=c-1$$ Then I substitute it into the Bernoulli's inequality, that is $$c^n\geq1+n(c-1)\geq 1+nc-n\geq nc+1$$ How ...
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### Proving a surjective function by given property

Suppose $f:E\rightarrow F$ and for any $A\subset F,A=f(f^{-1}(A))$. Show that f is surjective. What i have tried is $$\text{Let }y\in A$$ $$\{y\}\subset A$$ $$\{y\}= f(f^{-1}{\{y\})}$$ And i stuck ...
There are $5$ boxes containing $80$ candies. After taking $\frac{1}{5}$ of the candies in the first box and putting them in the seconf one, we take $\frac{1}{5}$ of the candies in the second box and ...