For posts seeking explanation or clarification of a specific step in a proof. "Please explain this proof" is off topic (too broad, missing context). Instead, the question must identify precisely which step in the proof requires explanation, and why so. This should not be the only tag for a question, and should not be used to circumvent site policies regarding duplicate questions.
Questions tagged [proof-explanation]
11824 questions
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1 answer
Measurable functions: $\sup f_n$ and $\limsup f_n$
I do not understand some passages of this proof. Can someone help me?
Let $X\ne\emptyset$ and $\mathcal{A}$ a $\sigma$-algebra on $X$. On $\mathbb{R}$ we consider the Borel $\sigma$-algebra $\mathcal{B}$.
Definition. An application
…
Jack J.
- 920
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1 answer
Proof Cartesian products and sets
Suppose A and B are subsets of a universal set E. Prove that
(E × E) \ (A × B) = ((E \ A) × E) ∪ (E × (E \ B)).
Is (E × E) \ (A × B) = ((E\A) ×(E\B)), any tips would be appreciated thanks.
Rivaldo
- 378
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1 answer
Confused by a proof from Solow book
I am working on the Solow book about how to do proofs.
Consider the problem of proving that, “If x and y are real numbers that $x^2 +6y^2 = 25$ and $y^2 +x = 3$, then $y = 2$.” In working forward the hypothesis, which of the following is not valid?…
siva
- 183
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1 answer
Given a sequence of functions $\{f_n\}$, $|f_n(x)|<\infty$ a.e. x. Why does the measure $m\{x\in [0,1]: |f_n(x)| \geq 1\}\leq 1$
I am reading a proof, but I couldn't understand the circled part. Can someone help me to understand why $m(F_1) \leq 1$?
user398843
- 1,771
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0 answers
On $\sum_{n \geq 1} \frac{f(kn)}{n^{s}}$
I am having a trouble understanding some equality in the book "The Theory of the Riemann zeta-function".
On pp. 9, I read
\[
\sum_{n \geq 1} \frac{\sigma_{a}(kn)}{n^{s}} = \prod_{p} \sum_{m \geq 0} \frac{\sigma_{a}(p^{l + m})}{p^{ms}},
\]
where $k…
Grown pains
- 166
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1 answer
Is it true that the nth derivative of this function is n factorial
so we were trying to show that the following functions derivative has $n-1$ roots using Rolle's theorem
Let $ a_1 < · · · < a_n \in \mathbb{R}$ and the function $f(x)$ be given by: $$f(x) = (x − a_1)· · ·(x − a_n)$$
So I applied Rolle’s Theorem for…
Yep
- 519
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1 answer
Doubts in proof of FTA (Fundamental Theorem of Arithmetic)
I have issues in understanding the proof of the F.T.A. by Wallace in the book titled: Groups, Rings and Fields, on page #66.
The issues are :
(i) It is stated that : "Then $p_1$ divides $q_1q_2...q_n$". I feel $q_n$is a typo, and should be…
jiten
- 4,524
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2 answers
Insight on why this equality may be true
I was messing around and noticed that it seems like $$\text{arctanh}(1/x) = \Re(\text{arctanh}(x))$$
Does anyone have any insight on why this is true, or how to prove it?
Ryan Goulden
- 803
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3 answers
Is it possible to have $x=2^N$ if $N$ is not integer and both $x$ and $N$ are rational numbers?
Is it possible to have $x=2^N$ if $N$ is not integer and both $x$ and $N$ are rational numbers?
If possible, give an example of $x$ and $N$. If not possible, explain why.
Charles Chou
- 21
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2 answers
Proving an inequality with multiple Variables
I was thinking of using induction, but i am not sure how to use induction when I have two variables $x$ and $y$. Can I just prove it for any $x$ and make $y$ a certain real number like $2$?
For real numbers $x, y, z$, and $x, y \geq 0$, prove that…
tskgreen
- 37
- 5
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2 answers
proof by contradiction with the well ordering property
I am having trouble understanding this proof that every integer from 2 onwards can expressed as a product of primes:
Assume the negation of the statement is true (proof by contradiction):
Negation of the statement: There exists an integer greater or…
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1 answer
Proving the contradiction between these two statements
Suppose that
$$\sqrt{2}+1 = \frac{m}{n}$$
where $m$ and $n$ are natural numbers with $m$ as small as possible. Deduce that we also have
$$\sqrt{2}+1 = \frac{n}{m−2n}$$
This is a contradiction. Why?
aNader
- 1
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2 answers
Prove the following sum identity by induction.
Could someone help me figure out where to begin with this proof by induction.
Prove by induction that $$\sum_{i=1}^n (i+1) = \frac{(n)(n+3)}{2}.$$
Edit: I have proven that the base case n=1 is true as it will give you 2=2. Next I should complete…
user489819
- 39
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3 answers
Divisibility Proof by Contradiction
For all y in the intergers and prime numbers x , if x divides y then x does not divide y+ 1
I understand you could prove this directly but apparently a proof by contradiction is easier (I just dont know how)
The basic form is to assume the…
rajteh
- 23
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4 answers
If $4\mid a+bc$ and $6\mid b+ac$ prove that $2\mid a^2-b^2$
If $4\mid a+bc$ and $6\mid b+ac$ prove that $2\mid a^2-b^2$
This is as far as I get:
$$a + bc = 4k\qquad\text{for some $k\in\Bbb Z$} \\b+ac =6l\qquad\text{for some $l\in\Bbb Z$}\\\implies (a^2-b^2)(1-c^2) = 16k^2 + 36l^2$$
Lmorj
- 109