For questions about the formulation of a proof, not about the mathematics behind it.

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2
votes
2answers
43 views

Proof verificication and question of rigour: $A$, $B$, connected implies union is connected

Don't mark this as duplicate. The other question does not help me figure out how rigorous MY proof is. Problem: Let $A$ and $B$ be connected subsets of a metric space and let $A\cap\overline{B}\neq\...
0
votes
1answer
44 views

Nice way to prove a limit.

I know how to prove the following limit $$\lim _{\epsilon \rightarrow 0} \frac{a^{\epsilon}-1}{\epsilon}=\ln(a)$$ But I am looking for a nice way to do it, a little elegant. Would you have one?
1
vote
2answers
57 views

Prove that the directrix-focus and focus-focus definitions are equivalent

(NOTE: This is my attempt at answering this question and this question, but I rewrote it in order to make it easier for me to solve. Also, I've made a YouTube video explaining this whole proof. Note ...
1
vote
3answers
63 views

Prove $\forall n \in \mathbb{N}$ where $ n \neq 1 , n + \frac{1}{n} > 2$ using completing the square.

I have got this far; I am only unable to understand how to finish the proof. $n>0 \implies n + 1/n > 0 \implies n + 1/n + 2 - 2 > 0 \implies {\big(\sqrt{n}+\frac{1}{\sqrt{n}}\big)}^2 - 2 >...
-4
votes
3answers
74 views

prove that the $5$th root of $r$ is irrational if $r$ is irrational [closed]

I am trying to learn mathematics for computer science in own efforts. I got this task to prove that $\sqrt[5]{r}$ is irrational, given that $r$ is irrational. Normally if I had to prove that $\sqrt{2}...
3
votes
1answer
124 views

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable?

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable? To clarify: the problem stated that the composition is well ...
1
vote
2answers
35 views

Show that a set is open if and only if each point in S is an interior point.

I am in a complex analysis class and have been asked to prove this. I know I have to prove both ways so. If a set is open then each point in $S$ is an interior point. Proof: Let $S$ be an open set,...
2
votes
1answer
145 views

How can I prove $\sqrt{2} ^{\sqrt{2}}$ is irrational? [duplicate]

I am learning proofs and a question was posed which asked us to prove that $\sqrt{2}^{\sqrt{2}}$ is irrational. They mentioned this - Hint: try using the log10 function... I tried my hand at the ...
4
votes
2answers
88 views

problem proving: $(1+q)(1+q^2)(1+q^4)…(1+q^{{2}^{n}}) = \frac{1-q^{{2}^{n+1}}}{1-q}$

I'm trying to prove this, and it is really frustrating, because it seems a really easy problem to prove, however, I'm having a little problem with exponents: $$(1+q)(1+q^2)(1+q^4)...(1+q^{{2}^{n}}) = ...
0
votes
1answer
59 views

Proof that $|S| \leq |T|$ if $S \subseteq T$.

Let $S$ and $T$ be sets. I am having trouble proving that if $S \subseteq T$, then $|S| \leq |T|$, where $|S|$ is the cardinality of $S$.
2
votes
6answers
102 views

Three positive numbers a, b, c satisfy $a^2 + b^2 = c^2$; is it necessarily true that there exists a right triangle with side lengths a,b and c?

If so, how could you go about constructing it? If not, why not? I am new to proofs and I was reading a book where they posed this question. I understand that if we are given any right triangle, the ...
0
votes
2answers
47 views

(Real Analysis) Topology: Prove $f(cl S)\subseteq clf(S)$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Show: $f(\overline{S})\subseteq \overline{f(S)}$ for $S\subseteq \mathbb{R}$ (Note: $\overline{S}$ denotes the closure of S; $\partial S$ ...
0
votes
2answers
54 views

(Rigor/Validity of Proof) Every sequence of reals in a compact set has a convergent subsequence

[ADDED/MODIFIED]: I began my proof with a compact set, but this was a wrong start. Although the comments are valid, I should've started with a bounded set. Because what I want to establish first is ...
0
votes
2answers
20 views

Strong Induction issue

I am trying to prove a statement using strong induction but I seem to be getting stuck. I don't know if did something wrong or I am just not recognizing an opportunity for factoring/how to factor ...
1
vote
2answers
65 views

Proof that $\overline{P(z)} = P(\overline{z})$ for polynomial $P$ with real coefficients

Let $$ a_0, a_1, a_2, a_3, \ldots , a_n \quad (n \ge 1)$$ denote real numbers, and let $z$ be any complex number. With the aid of $$ \overline {z_1 +z_2+ \ldots +z_n} = \overline z_1 +\overline z_2+ \...
0
votes
2answers
46 views

Easy proof the set of finite Set in countable is countable [duplicate]

Suppose I know a result that the set of finite sets in $\mathbb{N}$ is countable. Is there a very quick way to show that the set of finite sets in any $X$ countable is countable? Idea...two sets ...
0
votes
1answer
30 views

Show that every local homeomorphism is continuous and open therefore bijective local homeomorphism is a homeomorphism

Follow up on another question I asked recently: Topology: Show restriction of continuous function is continuous, and restriction of a homeomorphism is a homeomorphism Definition: Let $(X, \mathcal{...
2
votes
2answers
43 views

Topology: Show restriction of continuous function is continuous, and restriction of a homeomorphism is a homeomorphism

I need to prove two trivial results but I don't know how to work with restricted function and its inverse Consider the topological spaces $(X, \mathcal{T}), (Y, \mathcal{J})$ Claim 1: Let $f:X \...
0
votes
1answer
19 views

Proving $g(x) = E_x$.

I'm pretty new to mathematical proof and set theory and I'm having trouble proving this problem. I've started on it, but I don't know where to progress. Problem: Suppose that $f: A \rightarrow B$. ...
0
votes
2answers
46 views

Show that cuts are preserved under homeomorphism

Let $(X, \mathcal{T})$ be a topological space, assume that $X$ has no proper (not $X$ or $\varnothing$) clopen subset. Definition: A point $p \in X$ is a cut if $X \setminus\{p\}$ has a proper ...
3
votes
2answers
50 views

Football:What is the minimum number of points to guarantee qualification in a group of 4 teams?

In a group of 4 teams, each team plays 3 matches ( against the 3 other teams). A win gives a team 3 points, a draw 1 point, and a loss 0 points. In the end the top two teams of the group qualify. It ...
-1
votes
1answer
42 views

Show that a space is $T_1$ if all subset is the intersection of all open sets containing it

Following a question I asked yesterday Is this the correct definiton of $T_1$ space? I was left with a claim: $(X, \tau)$ is $T_1$ iff $\forall A \subseteq X, A = \bigcap\{U \subseteq X: U \in \...
1
vote
2answers
30 views

Show that $\mathcal{S}$ is a subbasis on $\mathbb{N}$

I was given a problem: (Edited) Show $$ S=\bigl\{S_p:p\in\mathbb{P}\bigr\}\cup\bigl\{\{1\}\bigr\} $$ where $\mathbb{P}$ is the set of prime numbers, and $S = \{n \in \mathbb{N}: n \text{ is a ...
0
votes
0answers
14 views

Propositional logic for a proof

I was able to prove the following proposition Suppose that $x > 0$ and that $y \in [0, 1] \cap S_x$. Then $$y \in [c(x), d(x)],$$ where $c(x)$ and $d(x)$ are two particular real valued ...
4
votes
6answers
122 views

Prove that $ 1+2q+3q^2+…+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $

Prove: $$ 1+2q+3q^2+...+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $$ Hypothesis: $$ F(x) = 1+2q+3q^2+...+xq^{x-1} = \frac{1-(x+1)q^x+xq^{x+1}}{(1-q)^2} $$ Proof: $$ P1 | F(x) = \frac{1-(...
0
votes
0answers
36 views

Convex, closed plane curve is a Jordan curve

The claim I'd like to prove or disprove is in the title. Here, convexity means every tangent line at any point on the curve has the whole curve in one half of it. The curve being closed means that it'...
2
votes
3answers
39 views

Principle of Mathematical induction proof

Prove that $2^n >n$ for all positive integer $n.$ I know this can be easily proved by using PMI Let $P(n): 2^n > n$ For $n = 1$ $$2^1 > 1.$$ Hence $P(1)$ is true. Assuming that $P(k)$ is ...
1
vote
1answer
24 views

Primitive = Non-negative + Irreducible + 1 Positive element on main diagonal

Can anyone provide me with the proof for the sufficient condition for a matrix to be primitive as described by the definition from planetmath.org? (http://planetmath.org/primitivematrix)
0
votes
2answers
72 views

$\sin 2x - \tan 2x = -\sin 2x\tan 2x$ trigonometric identity proof

I need to prove $$\sin 2x - \tan 2x = -\sin 2x\tan 2x$$ I tried simplifying $$ \sin 2x = 2\sin x\cos x;\quad \tan 2x = \frac{2\tan x}{1-\tan^2x}. $$ But it's so long and complicated that I ...
0
votes
1answer
21 views

Simpler way to do this proof for congruences of integers?

I am trying to do practice problems on proofs involving congruences of integers but I am stuck on half of the problem in that I am not able to consider a more simplified solution as opposed to brute ...
0
votes
3answers
33 views

If $(X, \mathcal{T})$ has a countable subbasis, then it has a countable basis

Given $(X, \mathcal{T})$ a topological space. Let $\mathcal{S}$ be a subbasis on $(X, \mathcal{T})$ Claim: If $\mathcal{S}$ is countable, then $\mathcal{T}$ has a countable basis $\mathcal{B}$ ...
2
votes
2answers
115 views

Prove that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +…+ \frac{n}{2^n} = 2 - \frac{n+2}{2^n} $

I need help with this exercise from the book What is mathematics? An Elementary Approach to Ideas and Methods. Basically I need to proove: $$\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +...+ \frac{n}{2^n}...
1
vote
2answers
65 views

How to prove the power set of the rationals is uncountable?

Recently a professor of mine remarked that the rational numbers make an "incomplete" field, because not every subsequence of rational numbers tends to another rational number - the easiest example ...
1
vote
0answers
18 views

What happen if we remove a newly created vertex resulted from an edge contraction of a 3-connected graph?

There is a little doubt along the way when I tried to prove to prove the following: Let $G\cdot e$ denote the contraction of edge $e$ in $G$. If $G$ does not have a Kuratowski subgraph and the ...
1
vote
1answer
77 views

Finite Union of Countable sets is countable

I have not studied the axiom of choice, I know how to prove that the union of two countable sets is countable and I want to use that a proceed by induction, but I'm not sure if my argument is okay. ...
0
votes
3answers
57 views

Why do you need to show A(1) before proving A(n) by induction? [duplicate]

My instructor stated that in order to have a valid proof by mathematical induction, you first have to show A(1) holds, and then assume A(n) to deduce A(n+2). Why is the first step necessary if we are ...
1
vote
2answers
48 views

Proof concerning regular space: there exist a closed set contained in any open set containing $x$

I was given a claim: Let $(X, \mathfrak{T})$ be a topological space. Then $X$ is a regular space iff $\forall x \in X, \forall U \in \mathfrak{T}$ s.t. $x \in U$, $\exists V$ such that $x \in ...
4
votes
2answers
46 views

My proof that $f[f^{-1}(D)] \subseteq D.$

I've just started studying formal proof and set theory, so it'll be really cool if someone can check out my proof for a pretty basic set theory problem. It'll be great if you can tell me if my proof ...
0
votes
1answer
30 views

Derivation of properties of Regular open sets.

I've been stuck on this question for quite a while and I would appreciate if someone could help me out. A is regular open iff $A=A^{{\bot\bot}}$ where $A^{\bot}$ = X - $\overline{A}$. $A^{\bot} = ...
0
votes
1answer
40 views

Proof that $a^x$ goes towards infinity as x goes towards infinity

I'm tasked to prove that $a^x \rightarrow \infty $ when $x \rightarrow \infty$ provided that (a > 1). I've found a very rigorous proof for this. But my question is, why can't it be logically realized ...
2
votes
8answers
234 views

Prove that $4$ divides $3^{2m+1} - 3$

Prove that $4$ divides $3^{2m+1} - 3$. By plugging in numbers I can see this is true, but I can't figure out a way to prove this, I was thinking maybe proving first that it is divisible by $2$, and ...
0
votes
1answer
28 views

How to show a continuous function from a space to a subspace is continuous from a space to the whole space?

Let $(X,\mathcal{T})$ and $(Y, \mathcal{J})$ be topological spaces. Let $W \subset Y$ be a subspace with its subspace topology. Show that if $f: X \to W$ is a continuous function, then $f: X \to ...
-1
votes
2answers
28 views

Proving by contradiction (6/9)

I have been given a statement that I need to prove using the contradiction method and I am just a little unsure of how to go about setting this up and executing. Here is the statement: If x is any ...
0
votes
3answers
44 views

Show that the closure of $A$ is the intersection of all closed sets containing $A$, topology proof needed

I want to show that given $(X, \mathcal{T})$, we define $\overline A = \{x \in X| \forall U \in \mathcal{T}, x \in U \implies U \cap A \neq \varnothing\}$ (definition of closure from Munkres), then ...
0
votes
0answers
13 views

Proof Function is Bounded/Unbounded

How can I prove that the function $$\sigma_i\left(t\right) = k_i\left[\left(a+b\left(T_i-t\right)\right)e^{-c\left(T_i-t\right)}+d\right]$$ is bounded/unbounded? Note: $\sigma_i\left(t\right)$ is ...
0
votes
0answers
36 views

Which is finer, co-countable topology or usual topology on $\mathbb{R}$?

We know that the usual topology is finer than co-finite topology on $\mathbb{R}$ How to show the usual topology is finer than co-finite topology on $\mathbb{R}$ And co-countable topology is (in ...
5
votes
3answers
176 views

How to integrate $\int_{-3}^3 (x^2-3)^{3} \,dx$ without expanding the polynomial?

How can I integrate: $$\int_{-3}^3 (x^2-3)^{3} \,dx,$$ neither expanding the polynomial nor using the relationship between integral and derivatives? I mean, there is a way to compute this integral ...
3
votes
4answers
87 views

Proof writing: $\sum_{n=1}^{\infty}| a_n|<\infty $ implies $\sum_{n=1}^{\infty} a_n^2<\infty $.

Let $\sum_{n=1}^{\infty} a_n $ be an absolutely converging series. By definition, this means $\sum_{n=1}^{\infty} \lvert a_n\rvert $ converges. We want to show that $\sum_{n=1}^{\infty} a^2_n $ ...
1
vote
0answers
49 views

Show $\mathbb{N}^{\{0,1\}}$ is uncountable with a hint

Let $\mathbb{N}^{\{0,1\}} :=\{f: \mathbb{N} \to \{0,1\}\}$ is uncountable I have never heard of the table approach, and all the proofs say uncountability of $\mathcal {P}(\mathbb{N})$ I have seen so ...
1
vote
4answers
41 views

Any n x n matrix A can be written as a sum A = B + C where B is symmetric and C is skew-symmetric.

How can i proof the following statement: "Any n x n matrix A can be written as a sum A = B + C where B is symmetric and C is skew-symmetric." i tried to work out the properties of a matrix to be ...