Tagged Questions

For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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0
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1answer
36 views

Number theory practice exam questions

Looking for some help for these two practice problems for my exam. I'll explain to you what I have so far and my ideas. So for this problem, I solved part (a) using induction, it wasnt too tricky. ...
0
votes
0answers
19 views

Proving the limit comparison test

I have the next attempt: Because $0<L< \infty$, we can find two positive and finite numbers, $m$ and $M$, such that $m<L<M$. Now, because $L = lim_{n\to \infty} \frac{a_{n}}{b_{n}}$ we ...
1
vote
0answers
40 views

Is my proof right of this result?

Suppose that: $\sum_{n=1}^{\infty}a_{n}$ converges absolutely and $\{b_n\}$ is bounded.Prove that $\sum_{n=1}^{\infty}a_{n}b_n$ converges absolutely. My attempt: Let $M$ be the upper bound of ...
2
votes
2answers
55 views

Verifying the convergence of a series.

I need to prove that the series $$\sum_{n=0}^{\infty}3^{-n}$$ converges and to find the limit. My attempt: We can express our series as: ...
-1
votes
1answer
19 views

Prove $|A| \le|C|$ for injection and surjective functions

$A$, $B$ and $C$ are finite sets with $F: A \to B$ a surjection and $G: B \to C$ an injection. Prove $|A| \le |C|$ I could prove it using examples, but not sure how to generally.
2
votes
2answers
42 views

Proof by Geometric Intuition may Fail!!

In a book on differential forms, I read, "After all, there were times when people took geometric intuition as proof, and later found that their intuition was wrong". I would like to see an ...
1
vote
1answer
56 views

Induction Proof with Combinations?

Show that for all $n\geq0$ $$\binom{n}{0}3^n+\binom{n}{1}3^{n-1}+\dotsc+ \binom{n}{n-1}3^{1}+\binom{n}{n} $$ $$= \binom{n}{0}5^n-\binom{n}{1}5^{n-1}+\binom{n}{2}5^{n-2}-\binom{n}{3}5^{n-3}+\dotsc ...
0
votes
1answer
48 views

Prove $\sin x=3x-2$ has only one real solution

Obviously you can draw a graph, but how would you prove this with calculus?
1
vote
3answers
39 views

Combinatorial Argument with Natural Numbers

Give a combinatorial argument to show that all natural numbers n ≥ k ≥ m c(n,k) * c(k,m) = c(n,m) * c((n-m),(k-m)) where c stands for combination.
1
vote
0answers
22 views

Prove that a set with n elements in union with an element not in the set has n+1 elements

Suppose $A$ has $n$ elements and suppose $a \notin A$, prove that $A \cup \{a\}$ has $n+1$ elements. I am pretty sure that I am supposed to use something with one to one and onto functions. It all ...
0
votes
1answer
23 views

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges. Also assume that the sequence is positive. lim sup$_{n} n^{2}a_{n} = 0$ means that for every $\epsilon$, ...
1
vote
1answer
22 views

Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges

Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges The solution proof goes like: lim inf$_{n} na_{n} > 1 \Rightarrow$ there exists an $N \in \mathbb{N}$ such ...
1
vote
3answers
22 views

Show that $7\mid 2^{3n} -1 \,\,\,\forall n\in \mathbb N^+$

Show that $7\mid 2^{3n} -1 \,\,\,\forall n\in \mathbb N^+$ Should I prove this by induction? If so, how should I go about it?
1
vote
0answers
9 views

Proofs regarding about all Second Derivative Test cases (Inconclusive & Single Variable)

This is how I would prove f''(c) > 0 that f(c) has local min and I would easily flip the inequalities and state a conclusion for f''(c) < 0 that f(c) has local max. Quick Proof for f''(c) > 0 ...
1
vote
1answer
53 views
+50

The fix points of the Möbius transformations are the eigenspace of a certain matrix.

Let $M=\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) \in GL_{2}(\mathbb{C})$ and we recall that the Möbius transformation attached to $M$ is the map: $z \to ...
-1
votes
0answers
42 views

Injective and Surjective Proofs

Suppose that $f:N\to A$ and $g:N\to B$ are bijective functions, and define a new function $h : N \to A \cup B$ by $$h(x)=\begin{cases}f(x/2)&\text{ if $x$ is even},\\g((x+1)/2)&\text{ if $x$ ...
0
votes
3answers
32 views

Proving quadratic inequalities?

I am trying to prove that $$e^{k+1} ≥ 3 + 3k + k^2$$ with, $$k>2$$ WhatI have done so far: What we are trying to prove is that $$e^n≥1+n+n^2$$ is a true statement. Since $n=3$ holds, this is ...
0
votes
1answer
17 views

Prove $|A| \leq |B|$ for $1-1$ function.

Prove $|A|\leq |B|$ if function $F:A\rightarrow B$ is a $1-1$ function. I wanted to know how to prove this out of curiosity. The help is appreciated.
-1
votes
4answers
43 views

Prove that $1+3+5+…+(2n-1)=n^2$ for every positive n integer [on hold]

Prove this statement using mathematical induction.
0
votes
0answers
32 views

Proving a Set's Equality

This is a homework question. The problem asks Construct an algebraic proof for the given statement. Cite a property form Theorem 6.2.2 for every step. Theorem ...
-1
votes
3answers
64 views

Find $\bigcap_{n=1}^\infty(0,1/n)=\emptyset$ [on hold]

I`ve tried this and is it true or completely not? Then how can I fix it? Proof: too wrong so I get it off
0
votes
1answer
16 views

Set difference bijection proof (stuck on injective)

So here is the question I'm working on So obviously I need to prove two things, that its both injective and surjective, however I'm trying to show it is injective and am currently stuck, here is what ...
0
votes
1answer
29 views

How to prove that a set is not totally ordered?

I know that a set to be totally ordered and for example $A,B \in P(X)$ must either be $A \le B$ or $B \le A$. And also $\le$ is equivalent to $\subset$ for sets. But I am not sure how I would prove ...
0
votes
0answers
49 views

Proving symmetry of metric (single linkage between clusters using arbitrary dissimilarity measure)

I am told to assume that our dissimilarity measure $d$ satisfies the properties required of it, what seems to be the definition of a metric: $d(x,y) \geq0 $ and $d(x,y)=0 \Longleftrightarrow x=y$ ...
0
votes
0answers
26 views

Show f is integrable and integral is C(b-a)

Let $f:[a,b]\to\Bbb R$ be as follows: $f(a)=A; f(b)=B$ and $f(x)=C$ for $a<x<b$. Show $f$ is integrable and the integral is $ C(b-a)$ Consider for real $a<b$ and real $A,B,C$, the function ...
1
vote
1answer
20 views

Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function.

Let $H$ and $K$ be normal subgroups of a group $G$, with $H \subseteq K$. Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function. [That is, if $Ha=Hb$, ...
1
vote
4answers
99 views

Proof that $f(x)=\frac{(\sin x)^3}{x}$ gets maximum in $(0,\infty)$.

I have this problem, and I got stuck in my proof Prove $f(x)=\dfrac{(\sin x)^3}{x}$ gets maximum in $(0,\infty)$. My Proof $$(1)\lim_{x \to 0+} \frac{(\sin x)^3}{x}= 0$$ $$(2)\lim_{x \to \infty} ...
0
votes
1answer
25 views

How to proove that smallest upper bound exists und it is cleary determined?

Let X be a set. Then a relation '$\le$' on $\mathcal P(X)$ is defined by: $A \le B :\Leftrightarrow A \subset B$ . Let $\mathcal A \subset \mathcal P(X)$. One set $B \in \mathcal P(X)$ for which ...
1
vote
1answer
23 views

Funny interconnection between a triangle and the ellipse inscribed

Le $p\in\Bbb R[X]$ be a 3rd degree polynomial. Suppose it has one real root and two complex conjugate roots: these three points forms a triangle in the complex plane. Consider the ellipse inscribed ...
0
votes
1answer
13 views

Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
0
votes
1answer
13 views

Understanding how to prove a bijection into three sets

I understand how to prove if there is a bijection from A onto B. However, say that there is a bijection from A onto B and a bijection from B onto C. How would I prove that that there is a bijection ...
0
votes
0answers
9 views

Prove the A x B lexicographical ordering is partially ordered

Is this proof? I think I may have the right ideas, but I'm not sure.
4
votes
0answers
44 views

Prove this congruence

Let $p$ be a prime of the form $4k+3$ and $m$ an even positive integer less than $p-1$. Prove that $$1^m+2^m+\cdots+\left(\frac{p-1}{2}\right)^m \equiv ...
3
votes
2answers
44 views

Prove that $4^{(p-1)/2}\equiv 1\pmod p$

If $p$ is a prime of the form $4k+3$, prove that $4^{\frac{p-1}{2}}\equiv 1\pmod p$. I was solving a problem and it came down to this. I have no idea how to prove it, I have tried. Any help would be ...
1
vote
1answer
28 views

Prove that it is transitive

Below is what I have so far. I'm pretty sure that it is transitive, but I'm not sure how to prove that it is. Prove that A is or isn't transitive.
2
votes
2answers
19 views

Help proving this recurrence relation?

Let $P_n$ be the number of strings of length n formed from letters A, B, C, E, O, that do not contain two consecutive consonants (that is, B or C). For example, AABOCA and BACOOEBO satisfy this ...
1
vote
0answers
34 views

Verify why a thing of if this proof is right

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some ...
22
votes
4answers
4k views

Why do proof authors use natural language sentences to write proofs?

I haven't read very many proofs. The majority of the ones that I've read, I've read in my first-year proofs textbook. Nevertheless, its first chapter expatiates on the proper use of English in ...
2
votes
3answers
44 views

How can I prove that the square of an even number ends in 0/4/6?

I am trying to prove that the last digit of the square of an even number is either 0, 4, or 6 but I'm completely lost and have no idea how to tackle this problem.
0
votes
1answer
31 views

How to prove distributive property of a determinant?

How to prove that $|A\cdot B| = |A|\cdot|B|$ where A and B are square matrices of the same size? P.S.: This proof is not mentioned in my textbook, nor was I able to find it on the web.
2
votes
1answer
33 views

Is there any good reason against referring to employed equations over the relation sign when establishing a new relation?

I need to write down a complicated proof for a paper, for which I need to employ equations that I established earlier for almost every new relation I show. I would consider it best for the reader, if ...
0
votes
2answers
55 views

$x,y$ are real, $x<y+\varepsilon$ with $\varepsilon$>0. How to prove $x \le y$? [on hold]

$x,y \in \mathbb R : x \lt y + \epsilon : \epsilon \gt 0$ Then prove $x \le y$ .
0
votes
0answers
36 views

Essential part to undestand a proof . [duplicate]

In the proof of the the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
0
votes
1answer
33 views

Question about a proof concerning abelian p-groups

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for ...
-1
votes
1answer
43 views

Prove that product of two continuous functions is continuous

Let $h,g$ be continuous functions Prove: The function $G:\mathbb R\to \mathbb R$ defined by $G(x)=h(x)\cdot g(x)$ is continuous
2
votes
2answers
71 views

A doubt with a part of a certain proof.

Well, in the proof of the following lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
0
votes
1answer
37 views

Why do a coset is isomorphic to a certain set.

I have encountered with the proof of the next lemma suppose G is a finite abelian p-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
0
votes
1answer
22 views

Determining equivalence classes

I have done (a), pretty straight forward. I understand an equivalence class as all the elements in the domain that map to the same result in the co-domain. For example in (mod 3), [|0|] would be the ...
0
votes
1answer
29 views

How to prove that lim sup $a_{n} \leq b$

Assume that $(a_{n})$ is a bounded sequence, prove that lim sup $a_{n} \leq b$ iff, for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ so that $n \geq N$ implies $a_{n} \leq b + \epsilon$ ...
1
vote
1answer
66 views

Let $f:[a,b]\to\Bbb R$ be as follows: $f(a)=A; f(b)=B$ and $f(x)=C$ for $a<x<b$. Show $f$ is integrable and the integral is $ C(b-a)$

Consider for real $a<b$ and real $A,B,C$, the function $f:[a,b] \to \mathbb R$ defined by $$f(x) = \begin{cases} A & x = a \\ B & x=b \\ C & a < x < b \end{cases}$$ I want to ...