# What is Proof: Definition + 999 Threads

A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

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1. ### I Monumental Proof Settles Geometric Langlands Conjecture

https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719 https://people.mpim-bonn.mpg.de/gaitsgde/GLC/ https://en.wikipedia.org/wiki/Langlands_program https://www.quantamagazine.org/what-is-the-langlands-program-20220601/...
2. ### I Tough lemma on locally finite refinement

Hello! I have some troubles diving in the proof of this lemma Lemma. Let ##S## be locally compact, Hausdorff and second countable. Then every open cover ##\lbrace U_\alpha \rbrace## of ##S## has a countable, locally finite refinement consisting of open sets with compact closures. Proof...
3. ### Why is the Jacobian for polar coordinates sometimes negative?

Proving this geometrically [1] gives ##J = r.## Why is the ##-r## one wrong? Why is ##(x, y) \rightarrow (\theta, r)## is different from ##(x, y) \rightarrow (r, \theta)##? Edit: In Paul's Notes [2] it seems like ##J## is always positive, but online says it can be negative... [1] The first...
4. ### A Question about existence of path-lifting property

I'm reading "Complex Made Simple" by David C. Ullrich and here i have a problem with the proof of a theorem: Theorem Suppose that ##p : X \to Y## is a covering map. If ##\gamma : [0,1] \to Y## is continuous, ##x_0 \in X## and ##p(x_0) = \gamma(0)## then there exists a unique continuous function...
5. ### Proving convergence of sequence from convergent subsequences

In the photos are two proof questions requiring proving convergence of sequence from convergent subsequences. Are my proofs for these two questions correct? Note in the first question I have already proved that f_n_k is both monotone and bounded Thanks a lot in advance!
6. M

### Proof given ##x < y < z## and a twice differentiable function

For this problem, My proof is Since ##f'## is increasing then ##x < y <z## which then ##f(x) < f(y) < f(z)## This is because, ##f''(t) \ge 0## for all t ## \rightarrow \int \frac{df'}{dt} dt \ge \int 0~dt = 0## for all t ##\rightarrow \int df' \geq 0## for all t ##f ' \geq 0## for all t...
7. M

### Differentiable function proof given ##f''(c) = 1##

For this problem, I'm confused by the implication from the antecedent ##0 < |x - c| < \delta## to the consequent. Should the consequent not be ##|f''(x) - f''(c)| < \frac{1}{2}## where ##\epsilon = \frac{1}{2}## (Since we are applying the definition of a limit for the first derivative curve)...
8. M

### Root test proof using Law of Algebra

For this problem, My solution is If ##c < 1##, then let a be a number such that ##c < a < 1 \implies c < a##. Thus for some natural number such that ##n \geq N## ##|x_n|^{\frac{1}{n}} < a## is the same as ## |x_n| < a^n## By Law of Algebra, one can take the summation of both sides to get...
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### Proving piecewise function is k-differentiable

For this problem, My solution is, ##F(x)=\left\{\begin{array}{ll} e^{-\frac{1}{x}} & \text { if } x>0 \\ 0 & \text { if } x \leq 0\end{array}\right.## The we differentiate both sub-function of the piecewise function. Note I assume differentiable since we are proving a result that the function...
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### Ratio test proof

For (a) and (b), Does someone please know how to prove this? I don't have any ideas where to start. Thanks!

34. ### B What theorems are available when using modulo arithmetic?

I'm looking for theorems related to using modulo arithmetic. As an example, if I apply a sequence of arithmetic operations to a given number to get an answer and then apply a modulo operation on the result to get a remainder in a given base. Wiil that be the same if I apply the modulo operation...

47. ### Induction with binomial coefficient

Hi, I'm having problems with the proof for the induction of the following problem: ##\sum\limits_{k=0}^{n} \frac{(-1)^k}{k+1} \binom{n}{k}=\frac{1}{n+1}## with ##n \in \mathbb{N}## I proceeded as follows: $$\sum\limits_{k=0}^{n+1} \frac{(-1)^k}{k+1} \binom{n+1}{k}=\frac{1}{n+2}$$...
48. ### Proving the Infimum and Supremum: A Short Guide for Scientists

Hi, I have problems with the proof for task a I started with the supremum first, but the proof for the infimum would go the same way. I used an epsilon neighborhood for the proof I then argued as follows that for ##b- \epsilon## the following holds ##b- \epsilon < b## and ##b- \epsilon \in...
49. ### Prove that ## 4\tan^{-1}\left[\dfrac{1}{5}\right]- \tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{4}##

I let, ## 4\tan^{-1}\left[\dfrac{1}{5}\right]- \tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{4}## ##\tan^{-1}\left[\dfrac{1}{5}\right]- \dfrac{1}{4}\tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{16}## Then i let, ##\tan^{-1}\left[\dfrac{1}{5}\right] = α ...
50. ### B Have I proved some part of Fermat's last theorem?

Have I proved Fermat last theorem? X^4 + Y^4 != Z^4 has been proved by Fermat that if X,Y,Z = integer numbers, the formular is fine. Set x=X^2, y=Y^2, z=Z^2, so x, y, z are (some) integer numbers based on X,Y,Z. x^4 + y^4 != z^4 //x, y, z are still integer, would be obey to Fermat's Fermat...