# Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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### How to show the inductive step of the strong induction?

Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 2, pg 341]. Problem: Use strong induction to show that all dominoes fall in an infinite arrangement of dominoes if ...
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### How to get $k^{k + 1} + k^k$ to equate $(k+1)^{k+1}$?

This is a problem from Discrete Mathematics and its Applications Let $P(n)$ be the statement that $n!<n^n$, where $n$ is an integer greater than $1$. $\quad(a)$ What is the ...
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### Reciprocal squares sum inequality [duplicate]

What is the easiest (preferably inductional) way without approximation of the sum_ to prove the following inequality: $\frac{1}{1^2}+\frac{1}{2^2} + \ldots +\frac{1}{n^2} \le 2 - \frac{1}{n}$
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### Contrapositive of this statement

Suppose $∀x ∈ R, ∃y ∈ R$, s.t. $∀z ∈ R.$ Consider the following statement: $$z > y \implies z > x + y$$ The contrapositive of this statement is: $$z ≤ x+y \implies z ≤ y$$ with the same ...
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### Defining prime numbers for proofs

In my discrete mathematics book under existence proofs it has Prove that there exists a prime $p$ such that $2^p -1$ is composite. It then goes on to say by trial and error we find $2^{11}-1$ ...
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### Show that $\lim (\sqrt{n^2+1)}-n) = 0$ [duplicate]

Can't use limit rules as $\sqrt{n^2+1}$ and n are not convergent sequences
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### Prove that if $\displaystyle \lim_{n\to \infty} x_n = x$ and if $x > 0$, there exists a natural number $M$ such that $x_n >0$ for all $n > M$.

Prove that if $\displaystyle \lim_{n\to \infty} x_n = x$ and if $x > 0$, then there exists a natural number $M$ such that $x_n >0$ for all $n > M$. Is this not just a proof of the ...
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### Definition of Ordinal (w/ Axiom of Regularity) (problem 37, page 208, Enderton's Elements of Set Theory)

Given the definition of an ordinal to be well-ordered by $\in$ and transitive, I am interested with proving the following: I know and understand the following which easily proves it, but uses the ...
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### Showing $(P\to Q)\land (Q\to R)\equiv (P\to R)\land[(P\leftrightarrow Q)\lor (R\leftrightarrow Q)]$ {without truth table}

Problem: Show $(P\to Q)\land (Q\to R)\equiv (P\to R)\land[(P\leftrightarrow Q)\lor (R\leftrightarrow Q)]$ Source: As was noted in the original post, this problem is from Daniel J. Velleman's book ...
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### Let $u,v \in \mathbb{R}^n, A = u\cdot v^T$ then prove $\|A\|_2 = \|u\|_2 \; \|v\|_2$

Let $u,v \in \mathbb{R}^n, A = u\cdot v^T$ then prove $\|A\|_2 = \|u\|_2 \; \|v\|_2$ How can I do this? I'm kinda stuck with this 2-norm of the matrix. If it would be $\|A\|_{\text{frob}}$ then ...
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### A question on proving the uniqueness of a mathematical object

When proving that there is a unique mathematical object that satisfies a particular condition, e.g., the inverse of an element of a group, is the intuition behind it the following? You assume that ...
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### Prove identity related to nths root of unity

If $1=z_0,z_1,...,z_{n-1}$ are nth roots of unity, prove that $$(z-z_1)(z-z_2)...(z-z_{n-1})=1+z+z^2+...+z^{n-1}$$ I don't know what is meant by the condition given. If I substitute ...
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### If $f:\mathbb{R} \to \mathbb{R}$ has two derivatives such that $f(0) =0$ and $f'(x) \leq f(x), \forall x,$ then $f\equiv 0 \ ?$

Suppose $f:\mathbb{R} \to \mathbb{R}$ has two derivatives such that $f(0) =0$ and $f'(x) \leq f(x), \forall x.$ Could anyone advise me how to prove/disprove that $f \equiv 0 \ ?$ Thank you.
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### Prove that every minimal edge cut in a simple connected graph $G(V,E)$ is even, then $G(V,E)$ has no odd degree vertex.

I need to prove/disprove the following statement -- If every minimal edge cut in a simple connected graph $G(V,E)$ is even, then $G(V,E)$ has no odd degree vertex. I am a bit confused about one of ...
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### why does $\varphi'(N)=0$ in this proof?

Fulton's book on page 105 defines $N$: Afterwards Fulton writes this solution for this lemma: I didn't understand why $\varphi'(N)=0$ Thanks
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### Any advantage when proving linear algebra statements without using bases?

I always felt that proofs in linear algebra that do not assume the existence of bases seem more elegant but is there also something mathematically more valuable about these proofs? I know that the ...
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### In the Collatz function, why does $2^k-1$ reach $3^k-1$ after $2k$ steps, and could it be used to find divergent trajectories?

If you start calculating the Collatz function from an integer of the form $2^k-1$, you will reach $3^k-1$ after $2k$ steps. So, it is possible to pick a starting value that continuously zig-zags ...
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### A question regarding a small calculation in the proof of a theorem.

I understand the theorem overall, and that the $\epsilon$ values are arbitrary. Notice how the values $\epsilon_1=\frac\epsilon{2M}$ and $\epsilon_2=\frac\epsilon{2(|t| + 1)}$ are just assumed, and ...
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### Two rows or two columns with the same number of plusses

I have tried drawn numerous tables in attempt to explain this and understand that the number of cells must be even however, I am not sure how to create this proof. I appreciate your support. Each ...
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### Show that $\sum_{n = 0}^{\infty} x^{n} = \frac{1}{1 - x}$ where $|x| < 1$ is not uniformly convergent

Show that $\sum_{n = 0}^{\infty} x^{n} = \frac{1}{1 - x}$ where $|x| < 1$ is not uniformly convergent My professor's proof is as follows: So we know that the radius of convergence is $R = 1$. Now ...
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Let $X$ be a nonempty set, and let $f$ and $g$ be defined on $X$ and ave bounded ranges in $\mathbb{R}$. Show that: $$\sup \{f(x) + g(x) : x \in\ X\} \leq \sup \{f(x) : x \in\ X\} + \sup \{g(x) : x ... 2answers 122 views ### Understanding how to prove limit theorems for sequences. How do you know what arbitrary value to choose for epsilon as in this document, and when should the triangle inequality be considered when writing your proof? 1answer 55 views ### Uniform convergence of \sum_{k=1}^{\infty}\frac{2^{k-1} x^{2^{k-1}-1}}{1+x^{2^{k-1}}} Does \sum_{k=1}^{\infty}\frac{2^{k-1} x^{2^{k-1}-1}}{1+x^{2^{k-1}}} converges uniformly. -1<x<1 I have tried to bound it by Weierstrass M-Test but haven't been successful. I have also tried ... 0answers 14 views ### Finding specific functions g_i in f(x)= \sum_{i=1}^{n} x^{i}g_i  Let f: \mathbb{R}^{n} \to \mathbb{R} be differentiable (may be not in C^{1}) and f(0)=0, show that there exists g_{i} : \mathbb{R}^{n} \to \mathbb{R} such that:$$f(x)= \sum_{i=1}^{n} ...
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My textbook says: To prove $A\Rightarrow B$ we have to lead $A \wedge \neg B$ to a contradiction. Does it imply, that $B\Rightarrow A$ would also be true? As far as I know $\wedge$ is commutative.
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### Proving that a function that calculates the cardinality of a given set is surjective on specified domain and codomain.

Define the set $A \subseteq\mathcal P(\mathbb N)$ as $$A = \{E \in\mathcal P(\mathbb N)\,:\, E\ \text{has a finite amount of elements}\}$$ Define a function $f: A\rightarrow \mathbb N$: ...
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### A decomposition of a differentiable function

this time I want to solve this problem: Let $f: \mathbb{R}^{n} \to \mathbb{R}$ be differentiable (may be not in $C^{1}$) and $f(0)=0$, show that there exists $g_{i} : \mathbb{R}^{n} \to \mathbb{R}$ ...
### Assuming $a_k + b_k = 1$ (Putnam 2003) [duplicate]
I do not understand as I wrote in a previous question: solution: I see that we can scale: $u_k$ but I do not understand why it is legal to say $a_k + b_k = 1$ what is $a_1 = 20$ and $b_1 = 1$ ...
Apparently, there exists a theorem, which says if a inequalities is homoegenous the terms can be multiplied by a scale $u_k$ like: (a_1 a_2 ... a_n)^{1/n} + (b_1 b_2 ... b_n)^{1/n} \le [(a_1 + ...