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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0
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2answers
22 views

Weird continuity proof

Let $I = [a,b]$ and let $f : I \to \Bbb R$ be a continuous function on $I$ such that for each $x$ in $I$ there exists $y$ in $I$ such that $| f(y)|\le | f(x)|/2$. Prove there exists a point $c$ in ...
0
votes
3answers
42 views

Prove that $A$ is countable.

Hi so I'm practicing for a exam and I need help to figure this proof out, Suppose $A\subseteq \mathbb R^+$, $b\in\mathbb R^+$, and for every list $a_1,a_2,\ldots,a_n$ of finitely many distinct ...
2
votes
1answer
29 views

Proving a set to be countably infinite.

I'm asked to decide and prove whether the set $\{\,x\in\mathbb{N}: |x-7|>|x|\,\}$ is finite, infinitely countable, or uncountable. I'm pretty certain it is infinitely countable. I say that since ...
8
votes
1answer
24 views

How to prove that it is possible to make rhombuses with any number of interior points?

I was given some square dot paper which can be found on this link: http://lrt.ednet.ns.ca/PD/BLM/pdf_files/dot_paper/sq_dot_1cm.pdf and was told to draw a few rhombuses with the vertices on the dots ...
1
vote
1answer
29 views

Riemann Sum proofs

If $f$ is Riemann integrable on $[a,b]$ and $\lvert f(x) \rvert$ $\le$ $M$ for all $x$ $\epsilon [a,b]$, show that: $\lvert \int_a^b f \rvert$ $\le$ $M(b-a)$ Just started learning Riemann sums ...
2
votes
2answers
33 views

Derivative Definition proofs

Let $f : \Bbb R \to \Bbb R$ be defined by $$f(x)=\begin{cases}x^2, & \text{if $x$ is rational} \\ 0, & \text{if $x$ is irrational} \end{cases}$$ Show that $f$ is differentiable at $x = 0$ and ...
0
votes
1answer
35 views

A proof about boundedness for continuous functions

Let $I := [a,b]$ and let $f : I \rightarrow \mathbb{R}$ be a continuous function such that $f(x) > 0$ for each $x$ in $I$. Prove that there exists a number $a > 0$ such that $f(x) \geq a$ for ...
0
votes
1answer
13 views

Boundedness Theorem for continuous functions on intervals

Just want to confirm this is a suitable proof: Assume $f$ is not bounded on $I$. So, for any $n \in \mathbb{N}$, $\lvert f(x)\rvert > n$. Since $I$ is bounded, $x_n$ is also bounded. By ...
-4
votes
4answers
24 views

Let $f,g$ be continuous from $\mathbb R$ to $\mathbb R$ [duplicate]

Let $f, g$ be continuous from $\mathbb R$ to $\mathbb R$, and suppose that $f(r) = g(r)$ for all rational numbers $r$. Is it true that $f(x) = g(x)$ for all $x \in \mathbb R$?
3
votes
2answers
322 views

Determine if the following is surjective

I need to determine if $f: \Bbb N\times\Bbb N \to \Bbb N$ such that $f(a,b) = a^b$ is a surjective (onto) function. My intuition is that it is but I don't know how to prove it. I don't even know how ...
2
votes
2answers
36 views

Prove that Set B is countable - Is this proof correct?

It seems that I have some issues with the rigor of this proof and I don't know what I'm doing wrong. Could someone tell me if this proof is correct and rigorous enough? Here's the question Prove ...
1
vote
2answers
14 views

Given a complete graph of n vertices Kn (has all possible edges – one edge between pair of vertices).

Given a complete graph of n vertices $K_n$ (has all possible edges – one edge between pair of vertices). Use counting to find a formula in $n$ for the number of edges in the graph. I know that the ...
1
vote
2answers
20 views

Cardinality of the union of two sets

I am having trouble attempting to prove the inequality $|X\cup Y| \le |X|+|Y|$. Here is my intuitive argument when we take the union of $X\cup Y$ if there are repeated elements then they are not ...
1
vote
2answers
41 views

How to prove countably infinite?

How do I prove the following set is countably infinite? $\{\frac{1}{n}: n\in\mathbb{Z}\setminus\{0\}\}$ I know that I can say this set is a subset of $\mathbb{Q}$, and that $\mathbb{Q}$ is infinite, ...
0
votes
1answer
18 views

Proving two Sets are Equivalent

If $A$ is a subset of the set of all functions $f:\mathbb{R} \to \mathbb{R}$ and let $g:\mathbb{R} \to \mathbb{R}$ be a bijective function. We use the notation $gAg^{−1}={g∘f∘g^{−1}:f∈A}$. Prove that ...
0
votes
0answers
34 views

Prove there are infinitely many polynomials $P$ such that $P<CH(P)$.

Prove there exists a constant $C(n) > 0$ such that for any $ξ ∈ [0, 1]$ there exists infinitely many polynomials $P(x) = a_nx^n + · · · + a_0 ∈ \mathbb{Z}[x]$ such that $|P(ξ)| < ...
0
votes
1answer
26 views

Prove by induction $n^{1/n} ≤ \frac{n+1}{2}$

The problem Prove by induction: $n^{1/n} ≤ \frac{n+1}{2}$ Attempt at solution I started off with the usual steps for an MI problem. We start with the $P_1$ case: for $P_1$, LHS = 1 and RHS = 1 ...
0
votes
1answer
17 views

Proof of Equivalence of Sets

If A is a subset of the set of all functions $f : \mathbb{R}\rightarrow \mathbb{R}$ and let $g:\mathbb{R}\rightarrow\mathbb{R}$ be a bijective function. We use the notation $gAg^{−1} = \{g\circ f\circ ...
0
votes
1answer
29 views

context-free languages operation closure

The following operation is defined on formal languages. $ operation1(L) = \lbrace w \ | \ wxy \in L, \ \forall x \forall y \ (|x|=|w|) \ \wedge (|y| = |w| ) \rbrace $ Prove that context-free ...
0
votes
0answers
25 views

Determining countably finite, finite, or uncountable

How can I determine whether the set of all differentiable functions is countably infinite, finite or uncountable? I want to say it is equivalent to $\mathbb{N}$, so it is countable? And I know it is ...
0
votes
1answer
21 views

What does it mean to prove that the addition of two countable sets is countable?

How Should I prove that $\mathbb{Q} + i\mathbb{Q}$ is a countable set? I've already proven that $\mathbb{Q}$ is countable.
0
votes
1answer
21 views

Proving equivalence of sets

How can I prove that the set $A=(0,1)$ is equivalent to the set $B=[1,\infty)$ ? I know I need to find a bijection from $A$ to $B$, but I'm not sure how to do so and prove that the function is ...
2
votes
1answer
34 views

Use Induction to Show $(1+a)^n \ge 1 + na$

If $a$ $\in$ $\mathbb R$ $\ni$ $a > -1$, then ($\forall n$ $\in$ $\mathbb R$) ($(1+a)^n \ge 1 + na$) My main concern is twofold: Firstly, I am concerned that constant $a$ in the proposition may ...
1
vote
0answers
55 views

Deduction method with a quantified statement

In this expression I am trying to prove is a valid argument using the deduction method that is using equivalences and rules of inference in a proof sequence. ...
1
vote
1answer
34 views

Let $R$ be a non-commutative ring. Show that if $R$ is simple and has 1, then $Z(R) = \{a \in R | ra = ar$ for all $r \in R \}$ is a field.

Let $R$ be a non-commutative ring. Show that if $R$ is simple and has 1, then $Z(R) = \{a \in R | ra = ar$ for all $r \in R \}$ is a field. I think what I need to do is to show that $Z(R)$ is simple ...
0
votes
0answers
34 views

How to prove this set P is countable? [duplicate]

Hi so I'm a beginner to proofs and these day's I'm studying infinite sets. I'm trying to figure out the proof for the following: Let P = {X$\in \mathscr{P}({\mathbb{Z}}^+)$| X is finite}. Prove ...
0
votes
0answers
21 views

Riemann Integrable Functions to prove $f(x) =0$ [duplicate]

Suppose that $f$ is continuous on $[a,b]$, that $f(x) \geq 0$ for all $x \in [a,b]$ and that $\int_a^b fdx = 0$. Prove that $f(x) = 0$ for all $x \in [a,b]$.
3
votes
1answer
24 views

Proving a recursive algorithm on a set is true

If I have an algorithm that returns the entry of a set with the largest value, how do I prove the algorithm is true mathematically? (I know I could just write tests for it.) I understand how to use ...
0
votes
1answer
22 views

Find a set A such that A∉Rngf

Let $f:\mathbb N\to \mathbb P(\mathbb N)$ be given by $$f(n)=\{m\in\mathbb N\mid 3m-10>n\}$$ I came up with $A=\{n\in\mathbb N\mid n\notin f(n)\}$, but I don't believe that it works because $6\in ...
0
votes
3answers
50 views

Proof By Induction $2^n \ge n^2$ for $n\ge4$

I am trying to prove the following, and here is what I have done: Can somebody help to complete this? $2^n \ge n^2$ for $n\ge 4$ $n=4$, LHS: $2^4 = 16$, RHS: $4^2=16$, $16=16$ Therefore TRUE Assume ...
0
votes
1answer
55 views

Proving that if the sequence $X_n$ converges to $x$, then ${X_n}^a$, where $a$ is a positive rational, converges to $x^a$.

I've been stuck on this problem for a while. I splitted a into $p/q$, so it would be $({X_n}^p)^{1/q}$, and I got the convergence of ${X_n}^p$ to be $x^p$ since it is just induction using the product ...
0
votes
1answer
36 views

Proving $2n-8<n^2-8n+14$ for all $n\geq 7$ by induction

For what values of the natural number $n$ is $2n-8 < n^2-8n+14$? (must use induction) I have determined that $n$ appears to work for all values except $n=4,5,6$. I was wondering if this proof ...
0
votes
2answers
25 views

Prove equation $(ad-bc)(a-c)^2 = (b-d)^3$, if polynomials has common root

$$\begin{split} W(x) &= x^3 + ax + b \wedge a,b \in \mathbb{R} &\wedge \mathbb{D}_W &= \mathbb{R}\\ G(x) &= x^3 + cx + d \wedge c,d \in \mathbb{R} &\wedge \mathbb{D}_G ...
1
vote
5answers
54 views

Mathematical Induction on a Subset of the Natural Numbers

I am given a strict inequality of the form $$ 2n - 8 < n^2-8n+14, $$ where $n$ belongs to the set of natural numbers $\mathbb{N}$ (in this case $n$ does not equal 0). I am asked, for what values ...
1
vote
1answer
52 views

What should I learn to increase my skill to find proof?

I know... reading lot of proofs and comments about them and working hard by myself on proving theorems are probably the only good solutions. But in the same time, it is not a solution at all because ...
4
votes
4answers
90 views

Prove $\sum_{i=2}^{n}\frac{1}{(n-1)n}$ = $\frac{(n-1)}{n}$ using induction.

I need to prove $\sum_{i=2}^{n}\frac{1}{(i-1)i}$ = $\frac{(n-1)}{n}$ using induction. I am getting stuck midway through the inductive step. Here is what I have: $\forall n\geq 2$, where ...
0
votes
1answer
15 views

Prove multivariable function is surjective?

I am a little confused on how to prove a multivariable function is surjective(onto). The function is $f: \mathbb N^2 \to \mathbb N$ such that $f(a,b) = a^b$ I tried thinking of a counter example but ...
-4
votes
1answer
41 views

Pythagoras theorem to find length of rope

A picture frame with a rope on top for hanging is $42$ cm across. It is hanging from a nail so that the vertical distance from the nail to the picture is $20$ cm. The strings is tied to the top ...
1
vote
2answers
22 views

Monotonous everywhere function

$f: \mathbb R \to \mathbb R,\forall x \in \mathbb R $ $\exists \delta \gt 0 : f$ is non-decreasing on $(x-\delta,x+\delta)$(I call that statement A). I need to prove that $f$ is non-decreasing on ...
0
votes
1answer
50 views

Show convergence of Power method

Given a symmetric positive definite matrix $A_0 \in R^{n \text{x} n}$ with Cholesky decomposition $A_0 = LL^T$. How can I show that $A_k$ converges to $diag(\lambda_1, ..., \lambda_n)$ where $A_k$ is ...
2
votes
2answers
32 views

Proof By Induction $n^2 > 3n$ where $n\ge 4$

I am trying to prove the following example, however I seem to be getting a little stuck: For $n\in\mathbb N$, $n\ge 4, n^2>3n$ What I have Done: Base Case:$ n=4$, LHS: $4^2 = 16$, RHS: $3\cdot 4 ...
1
vote
2answers
25 views

Proof by induction of the Inequality of Harmonic numbers: $H_{2^n} \ge 1+ \frac n2$

My question is, for the question below, in the inductive step, where does $\dfrac{1}{2^{(k+1)}}$ come from?And where does $2^k$ come from in the third last step?
1
vote
2answers
25 views

If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$.

Herstein 3.4.20: If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$. I don't understand why $\varphi$ needs ...
0
votes
0answers
14 views

Prove by induction: $E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$ Please just check what I've done

Prove by induction: $$E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$$ Let me show you what I've done. I think I'm right: $$n=1,$$ $$E[c_1U_1(X)] = c_1E[U_1(X)]$$ Okay so maybe this one looks ...
3
votes
2answers
28 views

How do I prove that $\forall \beta\in F(\alpha)\setminus F$ is transcendental?

Let $E/F$ be a field extension. Let $\alpha\in E$ be transcendental over $F$. Let $\beta\in F(\alpha)\setminus F$. Then, how do I prove that $\beta$ is transcendental over $F$? Here's how I tried: ...
0
votes
0answers
17 views

Prove multivariable function is injective?

I am a little confused on how to prove a multivariable function is injective(one to one). I know the process for single variables but got stuck sadly. The function f: N -> N such that f((a,b)) = a^b ...
2
votes
4answers
388 views

How do I write this proof formally?

How can I formally prove that $$\max\{\lvert x+y\rvert _i \} \leq \max\{ \lvert x_j \rvert \} + \max\{\lvert y_k \rvert \}$$ Where $x,y$ are the components of a $n$-vector with $1 \leq i,j,k \leq ...
1
vote
1answer
25 views

Dijkstra's Algorithm for Negative Weights.

Now the problem states that their is a graph $ G = (V,E) $ where some of the edges have negative weights while some of the edges have positive edges. Now the question is why won't Dijkstra's algorithm ...
5
votes
3answers
52 views

Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.

(Jones, p. 246) Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. This seems pretty easy to prove in the following way: Let $g_j$ be a ...
0
votes
0answers
28 views

Argument for finite solution of power Diophantione Equation.

Assume the equation $4x^3=y^2+3$ has infinite positive integer solution. If $x,y$ has general solution then it is clear that for any $x$(rational, integer), there is a $y$. It can be said there is a ...