# 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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### 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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### 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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### 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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### 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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### Examples of useful, insightful, and interesting hand-waving

I am really amused by the answers to this question on "Most harmful heuristic" posed on MathOverflow, from which I've benefited a lot. However, it seems to me that some hand-waving may be really ...
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### Trouble Understanding Proof Of Invariant Relationship

In part of a proof I am reading this is stated: $2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ) + (a_n + c_n )^2 + (b_n + d_n )^2 ≥ 2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ).$ (1) From this invariant inequality ...
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### Proving the Cone is Contractible: Is my Proof correct?

Is my answer/proof correct? Please help me make my proof more rigorous and accurate. I need everything to be absolutely clear and rigorous. Thank you. Question: Let $X$ be a topological space. The ...
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### A simple way to obtain $\prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^s}$

Let $p_1 <p_2 <\ldots <p_k <\ldots$ the increasing list in set $\mathbb{P}$ of all prime numbers . By Infinite geometric series $\sum_{k=0}^\infty r^k = \frac{1}{1-r}$ for all $s>1$ ...
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### How to show that a complex function have a branch in a domain

I've given as homework to show that the function $$f(z)=\sqrt{\frac{z+1}{z-1}}$$ has a branch on $G = \mathbb C \backslash [-1,1]$. I'm having a hard time in finding the way to approach this kind ...
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### Proving $P$ by proving $\neg Q$ and knowing $P\lor Q$

This may sound silly. I used to remember studying this in physics class and I thought of asking it in physics.stackexchange and then later I decided to ask it here itself. Let's say, under some ...
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### How to prove or statements

How do I prove statements of the following types: $A \text{ or } B \implies$ C $A \implies B \text{ or } C$ I don't know how to go about proving statements like this when they have "or". Can ...
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### How do I show $G_0$ and $G_1$ are conjugate subgroups? Please improve my answer.

Is my solution below correct? Please read through it and tell me if it seems complete or to make sense. Question: Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the ...
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### If $f$ s twice differentiable and satisfies the following constraints, prove $f'(0)>-\sqrt 2$

Let $f$ be a twice differentiable function on the open interval $(-1,1)$such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$and $f''(x) \le f(x)$, for all $x\ge 0$. Show that ...
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### Difference between bound and free variable

What is the difference between $\forall x (P(x)\implies Q(x))$ and $P(x)\implies Q(x)$ I know in the first one the variable x is bound but in the second one the variable is free. What are the ...
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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 ...
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### Counting candies in boxes

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 ...
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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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### Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...
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### Every Regular Language is a Context Free Language

How do I show that every regular language is a context-free language? I've been told to construct a Context-Free Grammar by Induction on the number of operators in the regular expression; but I'm ...
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