Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Questions with this tag are about the presentation of a mathematical proof. Questions might include:

  • Should I include [x-mathematical detail] at [y-part of this proof]?
  • Is the following a sufficient proof of [x-mathematical tidbit]?
  • I have written the following proof, could I somehow improve it, does it have good flow/can I improve readability?

But this tag is not for asking someone else to write a proof for you, or for how to answer some question. Questions such as: My professor asked me to prove the Pythagorean theorem and I don't know how to begin are not to have this tag.

This tag is intended for use along with other, more "mathematical" tags. A question about the writing of a proof in abstract algebra, for example, should have as well. This tag can be used along with the proof verification tag.

See here for a useful set of guidelines for writing a solution.

15776 questions
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Assume that n^2 is divisible by 3. Prove that n^2 is divisible by 9.

I'm trying to write a simple, informal proof to this problem. I know it would likely be simpler to tackle by showing a proof by contraposition, but I'm being asked specifically to write a proof by either contradiction or cases. In proof by…
Mariah Lynn
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Proof that if Wronskian is nonzero

Let $y_1$ and $y_2$ be real functions whose Wronskian is nonzero. Suppose that $Ay_1 + By_2= 0$. Prove that $A=B=0$ I have proved that $y_1, y_2 \neq 0$, if $A = 0 \rightarrow B =0$, and if $B = 0 \rightarrow A = 0$. However, how do I complete the…
rabito
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How can I prove $\sin(x^3)$ is injective in the interval $(-1,1)$?

I want to show that if $\sin(a^3) = \sin(b^3)$, then $a = b$. However, I can't take the arcsin of both sides without assuming it's injective, and it becomes a circular argument. Is there any other way I can prove that is injective from the…
Yoshi
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How can I prove that if Lim(xn) = x < Lim(yn) = y, then xn < yn?

I started out saying to fix e > 0 and that there exists N such that for all n >= N, |xn - x| < e and |yn - y| < e. But I don't know how to go from there, i.e. how can I utilize the x < y part to continue with my proof? Thank you!
TreausreDragon
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How to construct a natural deduction proof for the following question

Hi I have been trying this question for a while but I'm unable to construct a proof for the following question, these rules I have proved and these can be used in the proof any help would be much appreciated
Anon1325
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More ways of showing that $0.\overline 9=1$

It is known that $0.\overline 9=1$. I already have a couple of proofs. Proof 1: Let $x=0.\bar{9}\Rightarrow 10x=9.\bar{9} \Rightarrow 9x=9\Rightarrow x=1$ Proof 2: Let…
F. A. Mala
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How to show, that $z(y+1)=4(y+1)$ is $z=4$ rigorously?

For context: Theorem. There is a unique real number $x$ such that for every real number $y$, $xy+x-4=4y$. Proof. We show the existence by choosing $x=4$. Substituting it in gives. $xy+x-4=4y+4-4=4y$. Suppose there is another real number $z$ such…
Thomas Deniffel
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Let $f$ be continuous on $[0, 1]$ with $f(0) = f(1)$. Prove that there exists $c ∈ \left[0,\frac{1}{2}\right]$ such that $f(c) = f(c+\frac{1}{2})$.

Let $f$ be continuous on $[0, 1]$ with $f(0) = f(1)$. Prove that there exists $c ∈ \left[0,\frac{1}{2}\right]$ such that $f(c) = f\left(c+\frac{1}{2}\right)$. So, I know I'm supposed to use the Intermediate Value Theorem, and I can see generally how…
red
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Prove that there exists a real number $x$ such that $x^{177} + \frac{165}{1+x^8+\sin^2(x)} = 125$ using Intermediate Value Theorem.

Prove that there exists a real number $x$ such that $x^{177} + \frac{165}{1+x^8+\sin^2(x)} = 125$ using Intermediate Value Theorem. Uhhh I have no idea where to even start with this. Anything to give me an idea of what to do here would be great.
red
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Is this proof for this theorem about limit of a function correct?

Let $A\subset\mathbb{R}$ and consider a function $f:A\rightarrow\mathbb{R}$ and let $c\in\mathbb{R}$ be a cluster point of A. Theorem: The function $f$ does not have a limit at $c$ if and only if there exists a sequence $(x_n)$ in A with $x_n\ne c$…
Natasha J
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In an if and only if proof, why do you have to prove sufficiency?

This my be a simple question and I am missing something fundamental. When doing if and only if proofs, doesn't necessity imply sufficiency, so why does the 'if' have to be proven if the 'only if' is true?
AppleBottoms4
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Formality of writing proofs on, "If and only if," or similar statements.

Often in lectures when the lecturer proves a statement , $A \iff B$, they start off with writing ("$\Longrightarrow$") then proceeds to prove that $B$ follows from $A$ and then, ("$\Longleftarrow$") for the opposite. I understand this is usually for…
ASP
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Demonstrate $3+9+27+\cdots +3^{n}=(\frac{3}{2})\left ( 3^{n}-1 \right )$

Good day! I have this for this one, but I cannot get it. What am I doing wrong? $3+9+27+\cdots +3^{n}=(\cfrac{3}{2})\left ( 3^{n}-1 \right )$ $3^{1}=\left ( \cfrac{3}{2} \right )\left ( 3^{1} -1\right )$ $3=\left ( \cfrac{3}{2} \right )\left (…
lynneerwell
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$(a\neq -1\ \wedge\ b\neq -1)\implies (a+b+ab\neq -1)$ Using proof by contrapositive

Let A,B be two real numbers, using proof by contrapositive, show the following implication: $(a\neq -1\ \ and \ \ b\neq -1)\implies (a+b+ab\neq -1)$ I applied it's contrapositive $(a+b+ab=-1)\implies (a=-1\ or\ b = -1$) But I'm currently stuck in…
Cheeze
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