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

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0
votes
3answers
32 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
59 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
63 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
54 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
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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
45 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
29 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
200 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
27 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
2answers
25 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
174 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
86 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
43 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
3answers
30 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 ...
2
votes
0answers
63 views

Prove: The boundary of the set $\{(x,y,z) \in \mathbb{R}^3 | z > \sqrt{x^2+y^2}\}$ isn't a smooth manifold.

Prove: The boundary of the set $\{(x,y,z) \in \mathbb{R}^3 | z > \sqrt{x^2+y^2}\}$ isn't a smooth manifold. The boundary is defined by $z = \sqrt{x^2+y^2}$. I'm trying to think how to approach ...
0
votes
1answer
36 views

Prove $\forall n \in\mathbb{N}, (n|105 \wedge n|70) \implies 5|n$

I know the definition of divides into is $$a|b \equiv \exists a\in\mathbb{Z}, b = ac$$ however I'm not sure how to manipulate this to prove $$\forall n \in\mathbb{N}, (n|105 \wedge n|70) \implies 5|n$$...
1
vote
1answer
35 views

Proving $n < 2^n$ by Cantor's theorem

So we know Cantor's Theorem is of course. For any set $S$, the power set $P(S)$ has a strictly greater cardinality, $\iff \#S < \#P(S)$. We seek to prove $n < 2^n$ using this information. I ...
0
votes
3answers
33 views

Proving the u-substitution formula

Let $g: [a, b] \rightarrow [c,d] $ be continuously differentiable and $f: [c,d] \rightarrow \mathbb{R} $ continuous. Prove that 􏰀$\int_{a}^{b} f(g(x))g'(x) dx $ = $\int_{g(a)}^{g(b)} f(t)dt $ ...
4
votes
3answers
61 views

Disprove: $f\circ g = f \circ h \implies g=h$ for a surjective function $f$

I tried using a very specific counterexample here where I select a surjective function for which the compositions are equal but the functions within are not. This is probably off-base, but it's what ...
0
votes
1answer
17 views

Show two notions of dense are equivalent

This question follows from another one Topology proof: dense sets and no trivial intersection Show that given a topological space $(X, \mathcal{T}), D \subseteq X$ Then $D$ is dense iff $\...
4
votes
2answers
144 views

Proof of a statement about eigenvalues and eigenvectors.

How can i proof the following: Let $\mathbb L: V\rightarrow V $ be a linear mapping. Let $v_1,v_2,..,v_n$ non-zero eigenvectors with eigenvalues $c_1,c_2,..,c_n$ respectively, also let the ...
0
votes
1answer
17 views

How to proof that the set of all $X$ such that $X.A{\ge} c$ to some real number c is convex?

How can i proof the following statement: " Let $\mathrm A\in \mathbb R^{n}$ and $\mathrm c\in \mathbb R$, the set $\mathbb S$ of all elements belonging to $\mathbb R^{n}$ and satisfying the ...
3
votes
1answer
23 views

For $f: \mathbb{N} \rightarrow \mathbb{N}$ given by $f(x)=ax+b$, for what $a,b \in \mathbb{R}$ is $f$ a bijection?

I asked a similar question here. This question has different parameters however as you can see. For $f: \mathbb{N} \rightarrow \mathbb{N}$ given by $f(x)=ax+b$, for what $a,b \in \mathbb{R}$ is $f$ a ...
0
votes
1answer
35 views

If $f$ is injective, then $f(X\backslash A) = f(X) \backslash f(A)$

Given $f:X \to Y$ injective, $A \subseteq X$, then $f(X\backslash A) = f(X) \backslash f(A)$ I have spent a long time looking at this problem but I have not found a good way to approach this. Here ...
1
vote
2answers
30 views

Prove $f: A \rightarrow B$ is strictly injective, $\implies$ $f^{-1}$ is a function and dom $ f^{-1} \subset B$

The question I have about this proof is that, do I need to choose a specific function $f:A\rightarrow B$ that is not injective but surjective? Will I lose generality if I do? For instance, I was ...
1
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0answers
28 views

Show the Heaviside step function is continuous in $(\mathbb{R}, \mathcal{T}_\text{lower limit})$

Given $(\mathbb{R}, \mathcal{T}_\text{lower limit})$ where lower limit topology $\mathcal{T}_\text{lower limit} = \mathcal{T_\mathcal{B}}$ where $\mathcal{B} = \{[a,b) \subseteq \mathbb{R}, a < ...
2
votes
2answers
49 views

Prove $\forall n \geq 10, 2^n > n^3$

Prove $\forall n \geq 10, 2^n > n^3$ base case: $n = 10$ $2^{10} = 1024$ $10^3 = 1000$ $1024 > 1024$. So $P(k)$ holds for $k = n$. We seek to show $P(k+1)$ holds: We know $2^k > k^3$. ...
1
vote
1answer
36 views

Riemann Integrability defined by sequence of partitions

Prove that a bounded function $f$ is integrable on $[a, b]$ if an only if there exists a sequence of partitions $ \left(P_{n}\right)^{\infty}_{n=1} $ satisfying $$ \lim_{n\to\infty} [U(f, P_n) - L(f,...
0
votes
1answer
83 views

Obscure proof that $+$ and $\times$ are continuous?

I am looking for proof of $+$ and $\times$ are continuous operations without using the standard definition of continuity (1. $\epsilon-\delta$, or 2. preimage of open sets or 3. sequential ...
0
votes
0answers
28 views

How to show that $f(X \backslash f^{-1}(Y \backslash C)) \subseteq C$

Let $f: X \to Y$ be a continuous function, and that $C \subset Y$, then claim: $f(X \backslash f^{-1}(Y \backslash C)) \subseteq C$ Attempt: Immediately we run into a problem following: ...
2
votes
0answers
28 views

Powers of two in an infinite sequence of arbitrary integers

I'm not sure how to ask this properly and I haven't seen this problem anywhere yet, but I'm still interested if this can be (dis)proven or not. Consider a finite sequence of the numbers 1, 2, 4, and ...
4
votes
2answers
36 views

Prove $(A \cup B)' = A' \cap B'$

I would like some assistance in verifying this proof? (I understand the last conjecture about "symmetry" is probably shaky, but I just want to know if the first part is right since going backwards ...
7
votes
4answers
110 views

How to show that $f(x) = x^2$ is continuous using topological definition?

I am trying to show that simple continuous functions satisfy topological definition of continuity Recall given $(X, \mathcal{T}), (Y, \mathcal{J}), f$ is continuous if $f^{-1}(V) \in \mathcal{T}, \...
0
votes
1answer
25 views

Equivalence relations proof example?

Let $A$ = {$a,b,c$}. Give an example of a relation on $A$ that is anti-symmetric, reflexive on $A$ and symmetric. The first thing that one must do to proceed with this question is to first define ...
0
votes
1answer
28 views

Show $(\mathbb{R}, \tau_{co-countable})$ is not Hausdorff but every sequence converge to at most one point

Given $(\mathbb{R}, \tau_{co-countable})$, show that it is not Hausdorff but every sequence converges to at most one point. 1. If $(\mathbb{R}, \tau_{co-countable})$ is not Hausdorff, then $\...
1
vote
2answers
33 views

Is the intersection of a relation that is antisymmetric and a relation that is not antisymmetric, antisymmetric

Given a binary relation R,S on set A, assume that R is anti-symmetric. Show R intersection S is anti-symmetric. I started this proof by stating the definition of anti-symmetric with R which is $$ ∀...
1
vote
1answer
26 views

Equivalence relation and class, Proof.

The relation $P$ on $ℝ$ is defined by $xPy$ iff $x^2=y^2.$ (a) Prove that the the relation $P$ is an equivalence relation. (b) Describe the equivalence class of $3$ and $0$. In order to ...
0
votes
3answers
21 views

Nullity and an Isomorphism

I'm working on some introductory proofs in linear algebra, and I think that I could use some help on this particular problem. I want to prove that a linear surjective map $T: R \rightarrow W$ is an ...
9
votes
5answers
143 views

Prove that $2^n$ does not divide $n!$

I want to prove that $2^n$ does not divide $n!$. I was trying by induction and I'm confused about if what I'm doing is right. First I test it with $n=1$. In fact: $$2^1 \nmid 1!$$ So if i take the ...
2
votes
2answers
98 views

Show $\frac{1}{n}$ converges to $0$ using topological definition

I need to use the following definition to show that: $\frac{1}{n}$ converge to $0$ in $(\mathbb{R}, \mathcal{T}_{usual})$ and $(\mathbb{R}, \mathcal{T}_{lowerlimit})$ The defnition is: Given $(...
0
votes
1answer
24 views

Any additional constraint will increase the value of the maximin

In the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $325$, there is the comment "any additional constraint will increase the value of the maximin",...
0
votes
1answer
58 views

Prove that $\int_{a}^{b} f > 0$

I am asked to prove that $\int_{a}^{b} f > 0$. we are given that $f$ is continuous on $[a,b]$ $(\forall x \in [a,b]) \; f(x) \geq 0\; $ and $(\exists x_{0}) \in [a,b] \;s.t.\; f(x_{0}) >0$ my ...
0
votes
0answers
27 views

Proof Related to the Span in linear algebra

I'm working through a proof in my linear algebra textbook, and I think I am a little stuck. I am trying to prove that if $S$ is a non-empty set of vectors in a vector space $V$, the the set $W_s$ of ...