Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

10,082 questions
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Chebyshev Polynomials: Properties of Derivatives

Show that: $T_n'(x)$=$2n\sum_{k=0\\k+n~~odd}^{n-1}\frac{1}{c_k}T_k(x)$ $T_n''(x)$=$\sum_{k=0\\k+n even}^{n-2}\frac{1}{c_k}n(n^2-k^2)T_k(x)$ where $c_0=2$ and $c_n=1$ for $n\geq1$ I tried using the ...
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Elementary Proof on Perfect Squares.

The Proof I'm working on is: If $r \in \mathbb{N}$ is not a perfect square, then $\sqrt{r}$ is irrational.\ The farthest I've gotten is by proving by contradiction, assuming that $\sqrt{r}$ is ...
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Differentiability and continuity while partials have different conditions

The relevant things I read and will discuss are in this snapshot (from Folland Advanced calculus, and Wolfram Alpha, and this answer by zhw for an old question). Also, let me add two other links ...
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Number of points on the elliptic curve $\ y^2 = x^3 + 1$ [duplicate]

Consider the elliptic curve defined by $\ y^2 = x^3 + 1\$ over $\ \mathbb{Z}_p,\$ where $\ p \equiv 2 \pmod{3}\$ is prime. Prove that the number of points on the curve is exactly $\ p + 1.\$ Hint:...
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linear algebra eigenvalue proof question

Prove that a 2 × 2 matrix with just one eigenvalue (of multiplicity two) is diagonalizable if and only if it is already a diagonal matrix. So, by multiplicity two, I'm guessing it just means you have ...
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A question about space quotient and homeomorphism [duplicate]

Let $X$ and $Y$ be topological space and let be $\sim_X$ and $\sim_Y$ the corresponding equivalence relations on $X$ and $Y$. Let $f\colon X\to Y$ an homeomorphism, suppose we are in the following ...
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Proof by contradiction, status of initial assumption after the proof is complete. [duplicate]

First of all I'd like to say that I have looked for the answers to my specific question and have not found it in the existing topics. The question is fairly simple. Say, we need to prove statement P ...
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Combinatorics Proof of $\sum_{i=0}^n \sum_{j=0}^{i-1} j = {n+1 \choose 3}$

Proof of $\sum_{i=0}^n \sum_{j=0}^{i-1} j = {n+1 \choose 3}$ I am trying to generate a combinatorics proof of this identity, but have been stuck for hours. I've been trying to think of someway to ...
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Prove that matrix $A = \sum_k^{n}x_k\lambda_ky_k^T$

Prove that matrix $\ A = \sum_{k=1}^n {\bf x}_k \lambda_k {\bf y}_k^T.\$ If $\ {\bf x}_k\$ and $\ {\bf y}_k\$ are the corresponding eigenvector from the left and from the right of $\lambda_k$. Any ...
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Finding $\lim_{x \to \infty} \sqrt{x} c^x$ for $0<c<1$ [duplicate]

Is there a short way to prove that $\lim_{x \to \infty} \sqrt{x} c^x = 0$ for $0<c<1$? I tried using L'Hospital's rule and a few substitutions, and even if I was getting somewhere the proof was ...
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Showing that the simple continued fraction of $\sqrt{d}$ has period length 1 iff $d=a^2+1$

Given that I know if $d$ is an integer that $\sqrt{d}=[\alpha_0,\bar{\alpha_1},...\bar{\alpha_n},\bar{2\alpha_0}]$. I want to show that $\sqrt{d}$ has period length 1 if and only if $d=a^2+1$, for ...
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The units digit of a perfect square is 6. What are the possible values of the tens digit? [closed]

I know the answer to this already: the possible values of the tens digit are 1, 3, 5, 7, and 9. But I don't know how to prove it, can someone help please? Thanks!
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Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
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Proving the uniqueness of x=sqrt(r)

Given any $r \in \mathbb{R}_{>0}$, the number $\sqrt{r}$ is unique in the sense that, if $x$ is a positive real number such that $x^2 = r$, then $x = \sqrt{r}$ I would appreciate any nudge in the ...
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How to cite a theorem in the proof of another theorem?

I want to prove a theorem using the result of a well-known theorem. Should I write the well-known theorem as a lemma since it aids the proof of my theorem or as a theorem?
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Proving $N_2$ is not normal subgroup of $H_2$ if $\phi$ is not surjective

I am given that $\phi: H_1 \to H_2$ is a non-surjective group homomorphism and $\phi(N_1) = N_2$ where $N_1 \unlhd H_1$. How do I prove that $N_2$ may not be a normal subgroup of $H_2$? Attempt: ...
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How do I use induction to prove a claim of a recursive set definition?

The set X is defined as 12 ∈ X 15 ∈ X if x, y ∈ X, then x + y ∈ X if x, y ∈ X, then x − y ∈ X Claim: for every natural number n, 3n ∈ X I know I should induct on natural numbers that means my base ...
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$\cos(n\vartheta)=\frac{a_n}{3^n}$

I want to show, if i know that $\cos(\vartheta)=\frac{1}{3}$ than $\cos(n\vartheta)=\frac{a_n}{3^n}$ for $n\in \mathbb{N}$, where $a_n \in \mathbb{Z}$,$3 \nmid a_n$ My approach was to do it by ...
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Function $F$ is surjective if and only if $F$ is $1-1$ [duplicate]

While I was working on proofs of functions, the following claim occurred to me that I think it is correct but I could not prove it. Please note that the claim may not be correct since it is just my ...
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Question regarding partially ordered sets and the subset relation

Could someone look through my attempt at proving the following problem please? Let $(A, \preceq)$ be a POSET.For each $x,y \in A$,let $P_x=\{a \in A: a \preceq x\}$. Let $F$ be the family of sets ...
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Sequence\Limits Proof

Let $L = \lim_{k \rightarrow \infty}\limits x_k$. If $(x_k)_{k=0}^\infty$ is increasing, then $x_k \le L$ for all $k \ge 0$ Could anybody push me in the right direction? I've stared at this one for ...
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Prove the statement, For all integers $n ≥ 0$, $y_n = 3 * 2^n + 4^n$ where $y_0 = 4, y_1 = 10$ and when $n ≥ 2$, $y_n = 6y_{n-1} – 8y_{n-2}$

So approaching this problem For all integers $n ≥ 0$, $y_n = 3 * 2^n + 4^n$ where $y_0 = 4, y_1 = 10$ and when $n ≥ 2$, $y_n = 6y_{n-1} – 8y_{n-2}$ I realise that is probobly needs to be proved ...