Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [solution-verification]

For posts looking for feedback or verification of a proposed solution. "Is this proof correct?" or "where is the mistake?" is too broad or missing context. Instead, the question must identify precisely which step in the proof is in doubt, and why so. This should not be the only tag for a question, and should not be used to circumvent site policies regarding duplication.

This tag should be used when you have a proposed solution to a problem and have a concern or doubt about a specific step of the solution. A question with this tag should must reference a specific step of the solution and must include an explanation for why that step is in doubt. Further discussion on using this tag can be found in the Mathematics Meta questions (1) and (2).

Answers to questions tagged look first and foremost to check that the solution is right, and to comment upon the approach taken by the author of the question. If the proposed solution is wrong, a good answer would explain where or how mistakes were made, and possibly give or sketch a correction to the proposed method (when possible).

If the proposed solution is correct, an answer would ideally provide further evidence to back the answerer's opinion. These could include a clearer rewriting of the given argument, or careful consideration of unexpected subtle points, or generalizations of the proven result.

Answers should not duplicate existing content (use links if need be). Do not use solution-verification questions to repeat solutions that already exist on the site.

Users looking to write answers can find further discussion in this Mathematics Meta post.

23307 questions
115
votes
2 answers

$A$ and $B$ disjoint, $A$ compact, and $B$ closed implies there is positive distance between both sets.

Claim: Let $X$ be a metric space. If $A,B\subset X$ are disjoint, $A$ is compact, and $B$ is closed, then there is $\delta>0$ so that $ |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$. Proof. Assume the contrary. Let $\alpha_n\in…
Benji
  • 6,120
99
votes
14 answers

Formal proof for $(-1) \times (-1) = 1$

Is there a formal proof for $(-1) \times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed?
tvamsisai
  • 971
87
votes
6 answers

A new pythagorean proof

I am writing to you today because I believe I may have developed a new visual proof for the Pythagorean theorem. The proof is based on a geometric dissection method, equating the area of a larger composite square to the sum of its smaller…
kingyoon
  • 827
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69
votes
15 answers

Prove if $n^2$ is even, then $n$ is even.

I am just learning maths, and would like someone to verify my proof. Suppose $n$ is an integer, and that $n^2$ is even. If we add $n$ to $n^2$, we have $n^2 + n = n(n+1)$, and it follows that $n(n+1)$ is even. Since $n^2$ is even, $n$ is even. Is…
user79612
  • 575
68
votes
15 answers

Why do I get one extra wrong solution when solving $2-x=-\sqrt{x}$?

I'm trying to solve this equation: $$2-x=-\sqrt{x}$$ Multiply by $(-1)$: $$\sqrt{x}=x-2$$ power of $2$: $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$ But $x=1$ is not a correct solution to the original equation. Why…
TheLogicGuy
  • 1,036
59
votes
3 answers

A very odd resolution to an integral equation

Here is something I've found on the internet $$\begin{aligned} f-\int f&=1\\ \left(1-\int\right)f&=1\\ f&=\left(\frac1{1-\int}\right)1\\ &=\left(1+\int+\int\int+\dots\right)1\\ &=1+\int1+\int\int1+\dots\\ &= 1+x+\frac{x^2}2+\dots\\ &=…
Alma Arjuna
  • 6,521
57
votes
10 answers

Is a matrix $A$ with an eigenvalue of $0$ invertible?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof Suppose $A$ is square matrix and invertible and, for the sake of contradiction, let $0$ be an eigenvalue. Consider, $(A-\lambda…
Derrick J.
  • 571
55
votes
13 answers

If nine coins are tossed, what is the probability that the number of heads is even?

If nine coins are tossed, what is the probability that the number of heads is even? So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads. We have $n = 9$ trials, find the probability of each $k$ for $k = 0, 2, 4, 6, 8$ $n = 9, k =…
Stuy
  • 1,149
51
votes
9 answers

A game with $\delta$, $\epsilon$ and uniform continuity.

UPDATE: Bounty awarded, but it is still shady about what f) is. In Makarov's Selected Problems in Real Analysis there's this challenging problem: Describe the set of functions $f: \mathbb R \rightarrow \mathbb R$ having the following properties…
Gabriel Romon
  • 36,881
47
votes
5 answers

Prove that any two consecutive terms of the Fibonacci sequence are relatively prime

Prove that any two consecutive terms of the Fibonacci sequence are relatively prime. My attempt: We have $f_1 = 1, f_2 = 1, f_3 = 2, \dots$, so obviously $\gcd(f_1, f_2) = 1$. Suppose that $\gcd(f_n, f_{n+1}) = 1$; we will show that…
roxrook
  • 12,399
42
votes
9 answers

prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer

Prove that $\frac{(2n)!}{(n!)^2}$ is even $\forall n \in \mathbb{Z^+}$, i. e., a positive integer. My thought process: The numerator is the product of the first $n$ even numbers and the product of the first $n$ odd numbers; that is,$$(2n!) =…
hdtv1104
  • 621
42
votes
2 answers

Lipschitz Continuous $\Rightarrow$ Uniformly Continuous

The Question: Prove that if a function $f$ defined on $S \subseteq \mathbb R$ is Lipschitz continuous then $f$ is uniformly continuous on $S$. Definition. A function $f$ defined on a set $S \subseteq \mathbb R$ is said to be Lipschitz continuous…
Moderat
  • 4,627
42
votes
6 answers

Proof of the derivative of $\ln(x)$

I'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + h) - \ln(x)}{h} \\ &= \lim_{h\to0} \frac{\ln(\frac{x +…
rlms
  • 893
39
votes
8 answers

Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$

For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$. My attempt: Let $$\begin{align*} f_n(x) &= \frac{\ln\left(1-2 \left(a+\frac{1}{n}\right)\cos…
Stephen Dedalus
  • 1,806
39
votes
5 answers

Evaluating $\sum_{i=0}^2\sum_{j=0}^3(2i+3j)$

I've seen how nesting works with a simple $(i+j)$ but this problem below is tripping me up. It's either because of the multipliers or because they each start at zero but I get 60, and the answer I believe is 78. Not sure where I'm missing the last…
hobbes131
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