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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1answer
65 views

Convergence of a sequence by convergence of sub-subsequence

Suppose that $\{p_n\}_{n \in \mathbb{N}}$ is a sequence in a metric space $X$. Assuming that every subsequence of $\{p_n\}_{n \in \mathbb{N}}$ has itself a subsequence that converges, say, to $p$, ...
1
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3answers
41 views

Proving existence of a unique real number

I am working on the following question: For all $x \in \mathbb{R}$, $x \neq 6$, there exists a unique real number $y$ such that $xy+x=6y$. Now I have the existence part. That there exists a ...
0
votes
2answers
14 views

Proof of Injection and Surjection

I am having trouble proving the function f is injective and surjective. $f$ is a function from $\mathbb{Z}\times{Z} \to $\mathbb{Z}\times{Z}$ and $f(x,y) = (5x-y,x+y)$. I know it should be fairly ...
3
votes
2answers
54 views

Understanding a proof conceptually

Let's assume that $V$ and $W$ are vector spaces over a field $\mathbb{K}$, $\lambda\in\mathbb{K}$, $\lambda\neq0$. $S: V\rightarrow W$ and $T: W\rightarrow V$ are linear maps. Prove, that $\lambda$ ...
1
vote
1answer
35 views

Proving Inequality using Induction.

I am trying to prove the following statement: For every nonnegative integer $n$, $1+6n \le 7^n$. I did the base case where $n=0$ but am having trouble manipulating the inductive step. So far I ...
0
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2answers
19 views

Let $\text{A}$ be a nonsingular $\textit{n}\times\textit{n}$ matrix, and let $\textit{B}$ be a basis for $\mathbb{R}^n$

Show that $ B_1 = \{\textbf{Av}| \textbf{v} \in B\} $ is also a basis for $\mathbb{R}^n.$ I apologize for my informality, but I would really like some feedback as to whether I am using the correct ...
0
votes
3answers
68 views

If $ f $ is injective and $ g $ is injective, then $ f \circ g $ is surjective. [duplicate]

I can prove that if $ f $ and $ g $ are both injective, then $ f \circ g $ is injective, but I don’t know how to prove that $ f \circ g $ is surjective.
0
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1answer
45 views

Let $A = \{1- \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$.

Let $A = \{1 - \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$. I am supposed to figure out if this set is closed under certain topologies. I know that means I ...
0
votes
2answers
21 views

Suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$. I know this a true statement so now I need to ...
2
votes
2answers
71 views

Proving $\mathbb{Q} \times \mathbb{Z_2} \ncong \mathbb{Q}$

How do I prove $\mathbb{Q} \times \mathbb{Z_2} \ncong \mathbb{Q}$? I know that they are not isomorphic because for each element in $\mathbb{Q}$, say $\frac{a}{b}$, there are two corresponding elements ...
0
votes
2answers
23 views

proving a system with $A\vec{x} = \vec{0}=A^k\vec{x} $

The question: "Let $A\vec{x} = \vec{0}$ be a homogeneous system of $n$ linear equations in $n$ unknowns that has only the trivial solution. Show that if $k$ is any positive integer, then the ...
0
votes
1answer
22 views

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$.

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$. My definition of closure is "Let $(X, \mathfrak T)$ be ...
0
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0answers
21 views

Prove that h:$\mathbb{Z}\rightarrow\mathbb{O}$ where h(n)=2n-1 is bijective

I need to prove that h:$\mathbb{Z}\rightarrow\mathbb{O}$ where h(n)=2n-1 is bijective. I haven't done problems where $\mathbb{Z}\rightarrow\mathbb{O}$ and have seen no examples. I am only familiar ...
0
votes
1answer
49 views

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. If $Bd(A) = \emptyset$ then A =∅ or $A = X$ .

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. If $Bd(A) = \emptyset$ then $A = \emptyset$ or $A = X$. I am studying introduction to proofs and we have learned ...
2
votes
1answer
34 views

existence of multiplicity of roots [closed]

Im confuse..I read in an article that in dealing with polynomials, a quadratic equation can have either 2 real roots, 1 equal real root or 2 complex roots...but in dealing with random polynomials only ...
2
votes
1answer
20 views

Prove that $A\subseteq B$ if and only if $A^{C}\cup B=\mathscr{U}$

Prove that $A\subseteq B$ if and only if $A^{C}\cup B=\mathscr{U}$. I know we have to show that: if $A\subseteq B$ then $A^{C}\cup B=\mathscr{U}$ if $A^{C}\cup B=\mathscr{U}$ then $A\subseteq ...
2
votes
2answers
27 views

Help with Factorial Inequality Induction proof

So I'm asked to prove $3^n + n! \le (n+3)!$, $\forall \ n \in \mathbb N$ by induction. However I'm getting stuck in the induction step. What I have is: (n=1) $3^{(1)} + (1)! = 4$ and $((1) + 3)! = ...
2
votes
2answers
64 views

Is it ok to do this change of variable in integration: let $x = x - 1$

In integrals like $\int \sqrt{x-1}\,dx$, is it ok to make this change of variable in integration: "let $x = x - 1$"? It looks sketchy — like saying, let 5 = 4.
1
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1answer
31 views

Help understanding a proof about cardinal numbers

I was reading a proof about cardinal numbers, but I do not understand one step. The proof goes as follows: "Let $\beta$ be any ordinal, and for each ordinal $\alpha \lt \beta$, let $\kappa_{\alpha}, ...
0
votes
2answers
33 views

Prove a set is a subset of another. [duplicate]

I need to prove $A⊆B$ where A and B are defined as: ${A =\{x | x = 2n + 1}\}$ ${B =\{x | x = 2m - 21}\}$ where $n,m∈\mathbb{Z}$ I know that I need take an arbitrary element from A and show that it ...
0
votes
0answers
12 views

Proof the inverse image of set difference

I've the following exercise: Let $f:A \to B$ with $C,D \subseteq B$. Prove that $f^{-1}(D-C)=f^{-1}(D)-f^{-1}(C)$ For the proof, I've started from the definition of subsets a) $C,D \subseteq B ...
0
votes
1answer
8 views

To proof the difference images is a subset of their map difference sets

Let $A$ and $B$ sets, with $P,Q \subseteq A$ and let $f:A \to B$ 1) prove that $f(P)-f(Q) \subseteq f(P-Q)$ 2)Is it necessarily the case that $f(P-Q) \subseteq f(P)-f(Q)$? Give a proof or a ...
1
vote
4answers
31 views

Monotonicity of a fraction.

So I want to prove that the following fraction is monotone increasing, as a part of another proof, that's why I stumbled on: $$\frac{4^{n+1}}{2\sqrt{n+1}} \ge \frac{4^{n}}{2\sqrt{n}}$$ I know it's ...
2
votes
2answers
46 views

Limit proof check, show $f$ is bounded in a neighborhood of its limit point

edit: as lem has pointed out, the case where x=c is not handled. Could someone suggest an idea? Prove that if a function $f : A \to \mathbb{R} $ has a limit $l \in \mathbb{R} $ at $c \in L(A)$, then ...
0
votes
1answer
27 views

Barendregt's Substitution Lemma (lambda calculus)

I am struggling to put words on an idea used in Barendregt's Substitution Lemma's proof. (available here) The lemma states that: If x≠y and x not free in L and M, L are $\lambda$-terms: then ...
0
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2answers
35 views

Proof or a counterexample of a function

I have the following exercise, how can I proceed? Let $A$ and $B$ be sets, with $S \subset A$ and $f:A\to B$ a function, and $g:A\to B$ be an extension of $f\rvert_S$ to $A$. Does $g$ equal $f$? ...
0
votes
0answers
33 views

Exactly one solution to $ x\cdot a +y\cdot b = c $ with given $a,b \in \mathbb{N}\; c \in \mathbb{Z}$

how do I show that $ x\cdot a +y\cdot b = c $ with $a,b \in \mathbb{N}$ and $c \in \mathbb{Z}$ has exactly one solution in $\mathbb{Z}$. (There are 2 numbers $u,v \in \mathbb{Z}$ with ...
1
vote
0answers
30 views

Is the boundary of a set a subset of the limit points?

Let $(X, \mathfrak T)$ be topological space and suppose that $A$ is a subset of $X$. Then $Bd(A) \subseteq A'$. My definition of boundary: Let $(X,\mathfrak T)$ be a topological space and let $A ...
-1
votes
2answers
25 views

Prove the $Int(A) \subseteq A$. Using elements/ sets

Prove the $Int(A) \subseteq A$. My definition of interior is Let $(X, \mathfrak T)$ be a topological space and let $A \subset X$ is the set of all points $x \in X$ for which there exists an open set ...
0
votes
1answer
32 views

If A is a subset of a topological space, then $Bd(A) \subseteq Cl(A)$. Prove using elements/ sets.

If A is a subset of a topological space, then $Bd(A) \subseteq Cl(A)$. Prove. I know this statement is true. I am now trying to prove it. I am in a basic topology class and to do a lot of set ...
0
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1answer
30 views

If $A$ is a subset of a topological space, then $A' \subseteq A$ versus For any closed subset $A$ of a topological space, $A' \subseteq A$.

I need to determine which of the following are true and prove it... if it is false then I have to give a counterexample. If $A$ is a subset of a topological space, then $A' \subseteq A$ versus ...
1
vote
1answer
12 views

Invariant subspace (Proof)

How do I prove, that the eigenspaces of $T^n$ are invariant in regard to $T$, assuming T is an endomorphism in a real vector space V $(T: V\rightarrow V)$? That's how I started: Let $E_\lambda$ be ...
0
votes
1answer
35 views

How to go about a “not divisible by..” proof

I need to show the following proof: For any integer x, x^2 + 4 is not divisible by 3. I was trying proof by contraposition, but I do not believe that is the most efficient way to go about this. ...
6
votes
5answers
211 views

Proof writing: how to write a clear induction proof?

What is an effective way to write induction proofs? Essentially, are there any good examples or templates of induction proofs that may be helpful (for beginners, non-English-native students, etc.)? ...
0
votes
3answers
38 views

Prove that if the speed of a particle is constant, $\vec a$ is perpendicular to $\vec v$

My train of reasoning (well, not even that, more like "translating") so far: Let $C$ be some constant. $\|\vec {v(t)}\|=C$, so $\frac {d\|\vec {v(t)}\|} {dt}=0$. But, where to go from here? I don't ...
9
votes
3answers
107 views

role of definitions in proofs

Definitions are needed to define objects and such, however I am confused as to where definitions come from. I feel that they cannot be something that we arbitrarily define because simply saying ...
6
votes
5answers
49 views

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing.

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing. I do not have use derivatives, so I decided to apply the definition of being a strictly ...
0
votes
3answers
47 views

Suppose $A$ is a subset of $B$ and $B$ is a subset of $C$ and $A$ is equinumerous with $C$. Prove $B$ is equinumerous with $C$.

Definition I use: $A \sim B$ means $A$ is equinumerous with $B$ which means there is a $f\colon A \rightarrow B$ that is a bijection. My goal is to prove the following, Suppose $A \subseteq ...
0
votes
1answer
38 views

Prove that $\prec$ is irreflexive and transitive

Note: Definitions I use (Velleman's How To Prove It) If $A$ and $B$ are sets, then we will say that $B$ dominates $A$, and write $A \precsim B$, if there is a function $f: A \rightarrow B$ ...
0
votes
1answer
20 views

Let $(X, \mathfrak T)$ be a topological space and supposed that $A$ is a subset of $X$ then the Bd(A) is a closed set.

Let $(X, \mathfrak T)$ be a topological space and supposed that $A$ is a subset of $X$ then the Bd(A) is a closed set. I am in an introduction to proofs class. I have to decided if this is a true ...
1
vote
1answer
22 views

Let $A$ be a subset of $X$. Define $\mathfrak T = \{ U: A \subseteq U\} \cup \{\emptyset\}$. Then $\mathfrak T$ is a topology on $X$.

Let $A$ be a subset of $X$. Define $\mathfrak T = \{ U: A \subseteq U\} \cup \{\emptyset\}$. Then $\mathfrak T$ is a topology on $X$. I think this is a true statement and I therefore need to prove ...
1
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2answers
33 views

Prove using PMI that if $A$ is denumerable and $B$ is finite, then $A \cup B$ is denumerable.

This is what I have thus far: Claim: If $A$ is denumerable and $B$ is finite, then $A \cup B$ is denumerable. Proof. Suppose $A$ is denumerable and $B$ has $n$ elements and $B = \{b_1, b_2, b_3, ...
0
votes
2answers
57 views

Construction of Natural Numbers

I am trying to prove that the natural numbers can be constructed from the product of a power of $2$ and an odd number. For all $n \neq 0$ in the natural numbers, $n = (2k+1)(2^p)$, where $k$ and $p$ ...
0
votes
1answer
14 views

Question about choosing cases in a proof by cases

Prove that for every integer $n \ge 8$, there exist nonnegative integers $a$ and $b$ such that $n = 3a + 5b$. Proof: Let $n \in \mathbb Z$ with $n \ge 8$. Then $n = 3q$ where $q \ge 3, n = 3q + ...
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1answer
46 views

Function is identically zero almost everywhere

Prove that if $\int_E f d\mu = 0$ for some $f \ge 0$, then $f = 0$ almost everywhere. This is Execrise 1 in Chapter 11 of baby Rudin. My attempt: $\int_E f d\mu = 0 \implies$ sup { ${\int_E s ...
1
vote
2answers
23 views

shown that f is even

lets one function $f:\mathbb{R}-\{0\}\mapsto\mathbb{R}$ where $\mathbb{R}$ is the set of reals, lets $f$ such that $f\left(\frac{a}{b}\right)=f(a)-f(b)$ for every $a$ and $b$ in belonging to the ...
9
votes
3answers
1k views

Prove that there is no smallest positive real number

I have to prove the following: $$\text{Prove that there is no smallest positive real number}$$ Argument by contradiction Suppose there is a smallest positive real number. Let $x$ be the smallest ...
3
votes
1answer
70 views

Satisfiability proof of formulas with pure literals

Let $\varphi$ be any propositional formula in negation normal form (NNF). A literal $\ell$ is pure in a formula $\varphi$, if the complement of $\ell$, $\ell^c$, does not occur in $\varphi$, where ...
1
vote
2answers
34 views

Which of the properties, Reflexive, Irreflexive, Symmetric, Asymmetric, Antisymmetric, Transitive, Linear, does F satisfy?

Let $S={(n,m) ∶n,m∈Z^+}$. Define the relation F on S by ${(n,m),(i,j)}∈F$ if and only if $nj=mi$. In other words, let $F = {((n, m), (i, j)) ∈ S × S: nj = mi}$. Proof F is reflexive: Show that for ...
3
votes
1answer
27 views

Elementary question on set theory

Suppose $A \subset B$ then does this imply $B^{c} \subset A^{c}$? Here, $B^{c}$ denotes the complement of $B$. I have tried drawing Venn Diagrams and it seems obvious but is there a formal rigorous ...