# Questions tagged [proof-verification]

For questions concerning a specific proof or a specific solution, asking for verification, identifying errors, suggestions for improvement, etc. (You should not use this tag if the question does not contain a proposed proof/solution.)

21,922 questions
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### If $f:[0,1]\to \mathbb{C}$ be continuous with $f(0)=0$ and $f(1)=2$, then $|f(t_0)|=1$ for some $t_0 \in [0,1]$

Question: Let $T=\{z\in \mathbb{C}:|z|=1\}$ and $f:[0,1] \to \mathbb{C}$ be continuous with $f(0)=0$, $f(1)=2$. Show that there exists at least one $t_0$ in $[0,1]$ such that $f(t_0)$ is in $T$. ...
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### If $R$ is a right-Noetherian ring and the Jacobson radical satisfies the right Artin-Rees property, then $\bigcap^{\infty}_{n=1}\text{Jac}(R)^n = 0$

A (two-sided) ideal $I$ of a ring with identity $R$ has the right Artin-Rees property if for any right-ideal $E$ of $R$, there exists an integer $n\geq1$ such that $E\cap I^n\subseteq EI$. If $R$ ...
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### Counting integers with a least prime factor greater than $x$ in a sequence of $x$ consecutive integers.

It is well known from Sylvester-Schur that in any sequence of $x$ consecutive integers, there is always at least one integer divisible by a prime greater than $x$. I am interested in counting the ...
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### Check on a proof I saw on another thread: Metrizable Lindelöf spaces have a countable basis

I saw the following proof given of to the theorem below. I don't think the proof is correct, but I wasn't quite sure as it was given an up vote and thought I'd re post here to get some other opinions. ...
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### Uniform Convergence and continuity (Rudin 7.4)

This question has been asked for sequences, but I couldn't find it for series. Suppose $f(x) = \Sigma_{n=1} ^ \infty \frac{1}{1+n^2x}.$ $f(x)$ converges on $(0,\infty)$ and on $(-\infty, 0)$ ...
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### Formally deriving a vacuous truth from a definition involving conjoined implications

Definition of absolute value: $\forall x \in \mathbb{R}, (x \geq 0 \Longrightarrow |x| = x) \wedge (x < 0 \Longrightarrow |x| = -x)$ I want to use this definition in one of my proofs. So I ...
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### Question on the injectivity of a function.

Consider the map $f\colon\mathbb{S}^1\to\mathbb{S}^1$ defined as $f(z)=z^2$, where $\mathbb{S}^1$ is the unit circle, $$\mathbb{S}^1=\{z\in\mathbb{C}:|z|=1\}.$$ On $\mathbb{S}^1$ we define the ...
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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 $|u \cdot v| \leq ||u|| \ ||v||$ ...
### proof that $Y$ follows normal distribution
I'm new to probability and studying multivariate normal distribution. The one thing I don't understand is the linear transformation of multivariate normal distribution.If the $X$ follows Normal ...