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

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32 views

Prove intersection of these two sets is empty.

Define $x\sim y$ iff $x-y\in\mathbb{Q}$. Let $S = [0, 1)$. Define $A\subset S$ such that $A$ contains exactly one element of each class of equivalence of the relation $x\sim y$. Let $Q = \mathbb{Q}\...
17
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3answers
858 views

How can I answer this Putnam question more rigorously?

Given real numbers $a_0, a_1, ..., a_n$ such that $\dfrac {a_0}{1} + \dfrac {a_1}{2} + \cdots + \dfrac {a_n}{n+1}=0,$ prove that $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n=0$ has at least one real ...
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2answers
51 views

Proof of convergence of integral sequence using the definition of the limit [on hold]

"Let f(x) be a continuous, positive and decreasing function. Use the definition of the limit to prove that $$\int_{1}^{∞}f(x)dx$$ converges ⇔ the sequence $$a_{n}=\int_{1}^{n}f(x)dx$$ converges." ...
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0answers
37 views

Definition of function

When defining a function, mathematicians often write something like this: Let $f\colon \mathbb R\to \mathbb R$ be the function given by $x\mapsto x^2$. The purpose of this definition may be to ...
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1answer
30 views

How can I prove $\frac{m+a}{m+a+l+b}$ is between $\frac{m}{m+l}$ and $\frac{a}{a+b}$?

Let $m,a,l,b \in \mathbb{Z}^{+}$ How can I prove $\frac{m+a}{m+a+l+b}$ is between $\frac{m}{m+l}$ and $\frac{a}{a+b}$?
3
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1answer
59 views

Continuous functions in the product topology on $\Bbb{R}^{\Bbb{N}}$

I'm trying to prove the following statement: Let $(X, T )$ be a topological space, and let $f : X \rightarrow \Bbb{R^{\Bbb{N}}}$ be a function, where $\Bbb{R^{\Bbb{N}}}$ has the product topology. Let ...
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1answer
45 views

Elementary proofs involving inequalities

So the task of this exercise is to prove each statement. $\forall a \in$ $\mathbb R$: Prove that $a^2 \ge 0$ Does it suffice to say that $a^2 \gt 0$ or $a^2 = 0$, which means that $a \gt 0$ or $...
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0answers
63 views

When to use “:=”?

I know that "$:=$" is used in definitions. For example, one can write: $A\times B:=\{(a, b)\mid a\in A, b\in B\}$ But would you use "$:=$" in the following example: Let a function $f\colon A\...
0
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1answer
51 views

Understanding direct proofs and proofs by cases?

I'm reviewing my book Mathematical Proofs a Transition to Advanced Mathematics and looking to understand things at a deeper level. I will try to explain what I've considered so far in regards to this ...
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5answers
2k views

Why does proof by contrapositive make intuitive sense?

If you have two statements P and Q, and we say that P implies Q, that suggests that P contains Q. So if we have P, we must have Q because it is contained within P. This is my intuitive understanding ...
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2answers
59 views

Prove that ∀n≥1, (1/(1⋅3))+(1/(3⋅5))+(1/(5⋅7))+…+(1/(2n−1)(2n+1)) =( n/(2n+1))| [on hold]

Prove that ∀n≥1, (1/(1⋅3))+(1/(3⋅5))+(1/(5⋅7))+...+(1/(2n−1)(2n+1)) =( n/(2n+1))| So, I understand that the proof must display that (1/(2n−1)(2n+1) is equivalent to (1/(2n−1)(2n+1). Would I solve ...
0
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0answers
24 views

A partition of unity of a topological space

I have troubles in a little part of the following proposition. Let $(X,\tau)$ be a topological space and $\Im=\left\{U_{\alpha}\right\}_{\alpha \in I}$ an open cover of $X$. If $\Im$ has a locally ...
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0answers
38 views

Other simple proofs involving inequalities - when is it enough?

So the task of this exercise is to prove each statement. $\forall a \in$ $\mathbb R$: Prove that $a^2 \ge 0$ Does it suffice to say that $a^2 \gt 0$ or $a^2 = 0$, which means that $a \gt 0$ or $...
0
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1answer
21 views

Is collection of all functions I-convergent to a point form a ring?

$S$ be a set. $I$ is an ideal of $S.$ $X$ is a topological space. A function $$f: S\rightarrow X$$ is said to be $I$-convergent to a point $x\in X$ if $$f^{-1}(U)=\{ s\in S; f(s)\in U\}\in \mathscr F(...
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0answers
25 views

$\Bbb{R}^{\Bbb{N}}_{\square}$ is not ccc

Consider $\Bbb{Z}^{\Bbb{N}} \subseteq \Bbb{R}^{\Bbb{N}}$. The set $\Bbb{Z}^{\Bbb{N}}$ is uncountably infinite, since $|\Bbb{Z}^{\Bbb{N}}|$ = $|\Bbb{Z}|^{\Bbb{|N|}}$ = $\aleph_0^{\aleph_0}$ > $2^{\...
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0answers
32 views

Tietze Extension Theorem - How does the induction work?

I am reading a version of the Tietze Extension Theorem here: https://proofwiki.org/wiki/Tietze_Extension_Theorem There was a Lemma that says: And then it was repeatedly applied: How was the ...
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0answers
21 views

Style guide/typeface for handwritten mathematics

When writing math on graph paper, it's a small struggle to make my work as legible as possible and also use the page as efficiently as possible. I've read a little online about how latex typesets, ...
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5answers
855 views

APICS Mathematics Contest 1999: Prove $\sin^2(x+\alpha)+\sin^2(x+\beta)-2\cos(\alpha-\beta)\sin(x+\alpha)\sin(x+\beta)$ is a constant function of $x$

This is question 3 from the APICS Mathematics Competition paper of 1999: Prove that $$\sin^2(x+\alpha)+\sin^2(x+\beta)-2\cos(\alpha-\beta)\sin(x+\alpha)\sin(x+\beta)$$ is a constant function of $x$...
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3answers
26 views

When letting an element be an arbitrary member of a set $A$ do I have to account for $A =\emptyset$?

When letting an element be an arbitrary member of a set $A$ do I always have to account for $A =\emptyset$? E.g., if I wanted to prove (an incorrect) theorem $$ \forall A,B,C,D : A \times B \...
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3answers
35 views

Show that anti-metric space can only have one point

Let's define new object. Given $X$ a set: Let anti-metric be defined as: $b: X\times X \to \mathbb{R}$ such that: $b(x,y)\ge 0, \thinspace \forall x,y \in X$ $b(x,y)=0\Rightarrow x=y$ $b(x,y) = b(y,...
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3answers
45 views

What is the proper usage of $f: X \to Y$ and $f: \mathcal{P}(X) \to \mathcal{P}(Y)$ in proof writing.

I have read somewhere that suppose we are given a $$f: X \to Y$$ then $f$ is further associated with $$f: \mathcal{P}(X) \to \mathcal{P}(Y)$$ Does "associated" here means extended to a set valued ...
2
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2answers
33 views

The Greatest Number of Edges on a Bipartite Graph

Let $G$ be a bipartite graph on $p$ vertices. Find a formula in terms of $p$ that determines the greatest number of edges that $G$ could have. Prove that this formula is correct. Let $V$ be the set ...
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1answer
21 views

Show that $\bigcap_{\alpha} A_{\alpha}\times \bigcap_{\beta} B_{\beta}=\bigcap_{(\alpha,\beta)} A_{\alpha}\times B_{\beta}$

Show that $\bigcap_{\alpha} A_{\alpha}\times \bigcap_{\beta} B_{\beta}=\bigcap_{(\alpha,\beta)} A_{\alpha}\times B_{\beta}$ Surely this question is a duplicate but I dont know how to search these ...
2
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3answers
54 views

Proof related to Harmonic Progression

The question is as follows: Let $m_1<m_2<m_3<\cdots<m_k$ be postive integers such that $\frac{1}{m_1}$, $\frac{1}{m_2}$, $\frac{1}{m_3}$, $\cdots$, $\frac{1}{m_k}$ are in arithmetic ...
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0answers
17 views

How to prove in $r_1p_1 +r_2p_2 =u\gcd(p_1,p_2)$, $u$ is a uniformly random polynomial.

Hypothesis: All polynomials are defined over a finite field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit prime number). Assume we have two fixed polynomials $p_1$ and $p_2$ of ...
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0answers
40 views

Derivative of Exponential map on manifolds

I'm trying to compute the derivative of the map $f:\Sigma\times [0,\delta)\to M$ given by $$f(p,t)=\exp_p tN(p),$$ in $X\in T_p\Sigma$, where $(M^n,g)$ is a Riemannian manifold, $\Sigma\subset M$ a ...
0
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2answers
39 views

How to describe “the digits of n” mathematically where n is an integer?

Suppose n = 12345 The sum of the digits of n = 1 + 2 + 3 + 4 + 5 = 15 For example, in Python, we might isolate the digit 1 by writing n[0]. How would one represent the digits of n mathematically?
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0answers
13 views

proof of preimage of union and intersection of sets

I was learning to proof the following proposition "the inverse image of an intersection or union equals the intersection or union of the inverse image" following these two really good youtube videos: ...
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1answer
37 views

How to construct a proof once we intuit a solution

For any integer N, there is an integer P such that one of the following is true: N = 10P N = 10P + 1 N = 10P + 2 N = 10P + 3 N = 10P + 4 N = 10P + 5 N = 10P + 6 N = 10P + 7 N = 10P + 8 N = ...
0
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0answers
54 views

Challenging Series example [duplicate]

Let $\{x_n\}$ be a decreasing sequence such that the series of $x_n$ converge. Show that the limit as $n$ approaches infinity of $\{nx_n\}$ equals zero.
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1answer
34 views

Box topology is finer than the uniform topology on $\mathbb{R}^\mathbb{N}$

This time, I wish to show that the box topology is finer than the uniform topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ However, the problem here is that ...
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1answer
27 views

Uniform topology is finer than the product topology on $\mathbb{R}^\mathbb{N}$

I wish to show that the uniform topology is finer than the product topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ We know both spaces are metrizable: The ...
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3answers
41 views

Show $d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$ is a metric on $C[0,1]$

I am surprised that this question hasn't been asked on here I need to show that $$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$ is a metric on $C[0,1]$ Proof: As usual, positive ...
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3answers
34 views

How to show that any separable space is CCC

I thought I had the proof of this in my head, but it doesn't sound right on paper. Can someone see if my argument could be improved. Let $(X,\tau)$ be a topological space that is separable, then it ...
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1answer
35 views

Discrete Math Proof verification: products of floor

Determine if the following is true or false and provide a proof: $\forall x\in\mathbb{R},\exists y\in\mathbb{R}$ so that $\lfloor xy\rfloor = \lfloor x\rfloor \lfloor y\rfloor$ My attempt: -The ...
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2answers
19 views

Show that any metrizable space $X$ is regular

This is a quick follow up to another question Show that any metrizable space $X$ is Hausdorff Recall, a topological space $(X,\mathcal{T})$ is regular if we can separate any point $x$ from ...
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2answers
82 views

Proving the Baire Category Theorem from scratch, Stuck!

Theorem: (Baire) $(X,d)$ is a complete metric space then the intersection of countably many dense, open subsets in the metric topology $\mathcal{T}$ generated by $d$ is dense In other ...
2
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2answers
81 views

Are theorems like subroutines for math? [closed]

I've been developing more appetite for math just lately, as I study electromagnetics to deepen my understanding of electric circuits and devices. I'm finding that doing derivations as exercises helps ...
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2answers
46 views

Suppose X~Y, Prove that P(X) ~ P(Y)

My attempt: I imagined that if two sets are equivalent there would exist $ f:X→Y$ that is bijective. If I conceptually create P(X) and apply the function defined for the first equivalence relation to ...
2
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1answer
55 views

What does it mean in general to show something is well defined? [duplicate]

There is another post that addresses this but quickly fix the problem to be something in arthmetics, and in turn what it means for that arithematics problem to be well defined. I have never ...
2
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1answer
35 views

Show that any metrizable space $X$ is Hausdorff

I wish to show that any metrizable space $(X,\mathcal{T})$ is Hausdorff Proof attempt: Let $d$ be the metric that generates the topology on $X$. Pick two points $x,y \in X$, we wish to produce two ...
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0answers
34 views

Show that if $(X, \mathcal{T}), (Y, \mathcal{J})$ are both metrizable, then $X \times Y$ with product topology is metrizable

Let $(X, \mathcal{T}), (Y, \mathcal{J})$ be both metrizable, then Claim: $X \times Y$ with product topology is metrizable I have made an attempt at this question but the notation is ...
0
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1answer
26 views

How to write this proof in a purely formal way?

Problem. Let $\Lambda$ be an index set. Then show that $$\displaystyle\bigcup_{\alpha\in\Lambda}\left(\displaystyle\bigcup_{B\in\gamma_{\alpha}}B\right)=\displaystyle\bigcup_{B\in\gamma}B$$where $\...
0
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1answer
33 views

Proof idea: Let $(X,d)$ be a metric space, and $\rho$ be bounded metric, show that they will generate the same topology

Let $(X,d)$ be a metric space, $d$ generates the metric topology $\mathcal{T}$ via metric ball $B_\epsilon(x)$. Show that bounded metrics: $\rho_1(x,y) = \dfrac{d(x,y)}{1+d(x,y)}$ with ...
0
votes
1answer
58 views

Brute force way to show that $\rho(x,y) = \min\{1, d(x,y)\}$ is a metric

Following a hint in Short proof that $\rho^\prime(x,y) = \min\{1,\rho(x,y)\}$ is a metric I would like to use the brute force method to show that the standard bounded metric is a metric $$\rho(x,y) ...
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2answers
31 views

Trouble Proving that if $f : A \rightarrow B \text{ then } I_{B} \circ f=f$

Proving that if $f : A \rightarrow B \text{ then } I_{B} \circ f=f$ My problem with this question is that I do not know how one derives the theory in order to get the correct answer. I will ...
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1answer
35 views

A simple proof in the form of an inequality

Proof that for all $a, b$ are elements of $\mathbb{R}$ : $(a+b)^2\geq 4ab$. Does it satisfies after doing some simple arithmetic to say that $(a-b)^2\geq 0$? Or do I need to go over all the cases ...
0
votes
2answers
56 views

Prove $\lim _{ n\rightarrow \infty }{ { x }_{ n }^{ k } } ={ \left( \lim _{ n\rightarrow \infty }{ { x }_{ n } } \right) }^{ k }$

I'm trying to prove that the limit of the sequence $x_n^k$ is the same as the limit of $x_n$ all raised to the $k$th power. Prove $$\lim _{ n\rightarrow \infty }{ { x }_{ n }^{ k } } ={ \left( \lim ...
1
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0answers
29 views

How to calculate $\Delta$ in conic sections?

When learning conic section I learnt about $\Delta$. For any conic in general form : $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ Here $\Delta=abc +2fgh - af^2 - bg^2 -ch^2$ The conic is said to be ...
0
votes
1answer
41 views

Proving the Mean Value Theorem

So I'm working through some questions in my book and I don't understand how to finish out the attached problem. For the first 5 blanks here is what I got: Since $f$ is continuous on $[16,25]$ and ...