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.

learn more… | top users | synonyms

5
votes
3answers
234 views

Pythagorean theorem and its cause

I'm in high school, and one of my problems with geometry is the Pythagorean theorem. I'm very curious, and everything I learn, I ask "but why?". I've reached a point where I understand what the ...
1
vote
2answers
82 views

how to prove: $A=B$ iff $A\bigtriangleup B \subseteq C$

I am given this: $A=B$ iff $A\bigtriangleup B \subseteq C$. And $A\bigtriangleup B :=(A\setminus B)\cup(B\setminus A)$. I dont know how to prove this and I dont know where to start. please give me ...
1
vote
2answers
73 views

$T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$

Question Let $T:V \rightarrow V$ be a linear map. How do I prove that $T \circ T = T_0$ ( the zero linear map) iff $R(T) \subset N(T)$? Attempt \begin{eqnarray} T\circ ...
2
votes
1answer
70 views

If $x,y$ are elements of $\mathbb{R}$ and $x>0$ then there is a positive integer $n$ s.t. $nx > y$

Im reading a proof about this The proof is here. Let $A$ be the set of all $nx$, where $n$ runs through the positive integers. If $nx \le y$, then $y$ would be an upper bound of $A$. (start ...
7
votes
1answer
317 views

The genus of the Fermat curve $x^d+y^d+z^d=0$

I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation. Such curve is defined as the zero locus ...
-1
votes
1answer
61 views

Existence Proof: $T(v_i)=w_i$ for all $i=1,2,3,\dots,n$

Theorem to prove: Let $\{v_1,\dots,v_n\}$ be a linearly independent set in a finite-dimensional vector space $V$ and let $w_1,\dots,w_n$ be arbitrary vectors in a vector space $W$. Then there exists ...
4
votes
1answer
163 views

Hard-wiring a proof method in my head

There's s a kind of proof regularly used in linear algebra ( proving facts about Transformations, direct sums, basis, ... ) that i have definitely agreed with but still couldn't connect my intuitive ...
0
votes
1answer
82 views

Proof: Two circles have a most 2 intersections

I already prooved the statement here in general, but know I tried to proove it in an other way: I put $M_1$ on $(0/0)$ and the x-axis through $M_1$ and $M_2$. That simplifies the equatons for ...
0
votes
1answer
59 views

Proving recurrence relations

So, I initially proved the theorem that if $a != b^d$ and $n$ is a power of $b$, then $f(n) = C_1n^d + C_2n^{log_b a}$, where $C_1 = b^dc/(b^d − a)$ and $C_2 = f(1) + b^dc/(a − b^d )$. This is seen ...
5
votes
2answers
128 views

If $f$ and $g$ are continuous, prove $f\circ g$ is continuous.

Suppose that $(X,T)$, $(Y,U)$ and $(Z,V)$ are three topological spaces and that $g\colon X\to Y$ and $h\colon Y \to Z$ are continuous. Prove that $h\circ g\colon X \to Z$ is a continuous ...
0
votes
2answers
92 views

Proof pythagoras theorem with dot product + distance

I want to proof $d(A.B)^2=d(A,C)^2+d(B,C)^2$ for with $(\vec a-\vec c) \bullet (\vec b - \vec c)=0$. I applied the definitions of distance and got $d(A,B)^2=d(A,C)^2+d(B,C)^2 \Leftrightarrow ...
-4
votes
1answer
77 views

Transformation Existence Proof: A Call for Critique [duplicate]

QUESTION Prove that there exists a $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ ATTEMPTED ANSWER Let $V$ and $W$ be finite-dimensional vector spaces over $F$. Let ...
3
votes
2answers
115 views

Repeating Square Root Closed Form [duplicate]

I've been thinking about repeating square roots: $\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}$. I wrote a program on my calculator to do it $n$ times and I found that, if $x = y^2 - y$ then ...
0
votes
1answer
90 views

Explanation of proof common divisor divides gcd needed

I have been given a proof that uses the gcd to show that there is no biggest prime. However (coming from a less mathematical background) I am having trouble actually understanding what it means and ...
2
votes
1answer
63 views

Proofs regarding diagonalization of matrices

I'm trying to prove that a unitary matrix can be diagonalized using an orthonormal basis of eigenvectors, and that the eigenvalues are on the unit circle. So far, I have been able to show that if A is ...
3
votes
2answers
729 views

Solving two simultaneous recurrence relations

If we have the two recurrence relations $$a_n = 3a_{n-1} + 2b_{n-1}$$ $$b_n = a_{n-1} + 2b_{n-1}$$ with $a_0 = 1$ and $b_0 = 2$. My solution is that we first add two equations and assume that $f_n = ...
3
votes
1answer
49 views

Using a particular image to justify a (specific) trig integral equality.

I would like to include the following string of equalities in a paper: $$\sin ^2(x) + \cos ^2(x) = 1$$ $$\int _0^{\dfrac{\pi}{2}} \sin ^2 (x)dx + \int_0^{\dfrac{\pi}{2}} \cos ^2 (x)dx = ...
3
votes
1answer
118 views

Is my proof correct? $p_1p_2p_3\cdots p_n+1)$ cannot be the square of an integer

Prove that $p_1p_2p_3\cdots p_n+1$, where $p_n$ is the $n^{th}$ prime, cannot be the square of an integer. Let $p_1p_2p_3\cdots p_n+1=Q$ and assume it is the square of an integer, so ...
6
votes
2answers
165 views

Discreteness of eigenvalues for certain operators - can this approach be made rigorous?

I was idly thinking about why one might naïvely expect a discrete spectrum of eigenvalues for a linear operator $L$ when I dreamt up the following argument (which I expect isn't new instead - ...
1
vote
0answers
53 views

Visual/intuitive proof of why $\sum k^3 = (\sum k)^2$, where $k$ goes from 1 to $n$? [duplicate]

I understand that one could prove this by first proving the analytic expressions of the sigma terms through induction, and then square the $\displaystyle\sum_{k=1}^n k$ term to show LHS = RHS. Are ...
2
votes
3answers
231 views

is this a foolish way to do proofs?

When I'm asked something like "show X is equal to Y", I first try to manipulate what I know (X) into the result (Y). A lot of the time, I do not investigate the result I'm trying to conclude with. I ...
1
vote
2answers
1k views

A one-to-one function from a finite set to itself is onto - how to prove by induction?

I'm not sure if I can do this without knowing what f actually is? Let $X$ be a finite set with $n$ elements and $f: X \rightarrow X$ a one-to-one function. Prove by induction that $f$ is an onto ...
5
votes
3answers
2k views

Prove the following using induction on n (matrices)

Prove the following using induction on n: $$\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}^{n} = \begin{pmatrix} n+1 & n \\ -n & -n+1 \end{pmatrix}$$ I know that multiplication of ...
1
vote
2answers
178 views

Showing a contraction without a fixed point

Suppose $f: [1, \infty) \to [1, \infty]$ defined by $f(x) = x + \frac{1}{x}$ for all $x \geq 1$. I want to prove that: \begin{equation} |f(x)-f(y)| < |x-y| \end{equation} except when $x=y$, but ...
2
votes
2answers
294 views

Induction proof [ little-o notation ]

I have to prove that $ 2^n = o(n!) $, that is, $ \forall c \gt 0 \quad \exists$ $ n_0 \in \mathbb N$ such that $ \forall n \ge n_0$ we have $ 2^n \lt c.n! $ Well, this is what I did so far: First I ...
2
votes
1answer
53 views

Show that exists $x$ and $y$ such that $P(x)*P(y) < 0 \ ; x,y\in \Bbb C$

I would like to ask how could I perform following proof: Prove(show) that exists such $x$ and $y$ that $P(x)*P(y) < 0$. Where $P(x)$ and $P(y)$ are polynomials. $\forall\ x,y\in \Bbb C$
1
vote
2answers
244 views

Prove that a greedy algorithm selects the maximum number of programs

This is a homework problem. Intuitively, I know it to be true, because the largest group of programs (say, $j$ programs) must be composed of the smallest $j$ programs. But how to go about formally ...
4
votes
4answers
209 views

Combinatorial Proof

I have trouble coming up with combinatorial proofs. How would you justify this equality? $$ n\binom {n-1}{k-1} = k \binom nk $$
2
votes
2answers
179 views

$T:P_n(F) \rightarrow F$ PROOF OUTLINE

I'd like some heavy critique if you don't mind. See here for more details. Let $S=\{f \in P_n(F) : f(1)=0\}$. Clearly, the polynomial $f(x)=0 \in S$ because $f(c)=0$ for any choice of $c\in F$. To ...
1
vote
0answers
116 views

Determining Complete Metric Spaces

I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$ My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that ...
1
vote
3answers
76 views

Can someone check the solution to this recurrence relation?

Here's the recurrence relation: $a_n = 4a_{n−1} − 3a_{n−2} + 2^n + n + 3$ with $a_0 = 1$ and $a_1 = 4$ Here's the solution:Write: $$ a_{n + 2} = 4 a_{n + 1} - 3 a_n + 2^n + n + 3 \quad a_0 = 1, a_1 = ...
0
votes
2answers
58 views

Finding this solution to a recurrence relation

So, I know that the recurrence relation $a_n = 4a_{n−1} − 3a_{n−2} + 2^n + n + 3$ with $a_0 = 1$ and $a_1 = 4$ has the solution of $a_n = -4(2^n) - n^2 / 4 - 5n / 2 + 1/8 + (39/8)(3^n)$. I just ...
1
vote
3answers
60 views

Prove that $\frac{1}{\sqrt{2n-1}}-\frac{1}{2n}\geq \frac{1}{2n}$ for $n = 1, 2, 3,…$

Prove that $\frac{1}{\sqrt{2n-1}}-\frac{1}{2n}\geq \frac{1}{2n}$ for $n = 1, 2, 3,...$ This is required to prove that the series $1-\frac{1}{2}+\frac{1}{\sqrt{3}}-\frac{1}{4}$ is divergent, but I ...
2
votes
1answer
475 views

Limit of Binomial distribution

In showing us that Binomial distribution: $$B_{N,p}(n) := \binom {N}{n} p^n(1-p)^{N-n}$$ tends to Poisson's: $$P_ \lambda (n) = \dfrac {\lambda ^n}{n!}e^{-\lambda}$$where I guess lambda should be ...
0
votes
1answer
29 views

How to show all solutions for a particular recurrence solution

I've found that the recurrence relation $a_n = 4_{an−1} − 4a_{n−2} + (n + 1)2^n$ has the solution of $an = 2^n(p_0 + p_1n + n^2 + n^3/6)$. I'm just trying to understand the steps necessary to solve ...
1
vote
1answer
71 views

Diagrammatic Representations: $\dim(Skew_{n\times n}(\mathbb{R}))+\dim(Sym_{n\times n}(\mathbb{R})) = \dim(M_{n\times n}(\mathbb{R}))$

SEE AUTHOR'S ANSWER BELOW So I'm trying to derive the dimensions of both $Skew_{n\times n}(\mathbb{R})$ and $Sym_{n\times n}(\mathbb{R})$. I know that $\dim(M_{n\times n}(\mathbb{R}))=n^2$, but I ...
1
vote
2answers
70 views

Find the recurrence solution of this relation

How would we find the solution of the recurrence relation: $a_n = 2a_{n−1} + 3 · 2^n$ ? After trying it, I've found it to be $a_n = 2^{n-1} (c_1 + 6n)$ Not sure if this is right.. Thanks!
1
vote
1answer
44 views

On the Dimensionality of Space: An Elementary Analysis

The below theorem I am to prove. Perhaps you have a critique... Theorem 2.4 Let $W_1$ and $W_2$ be two subspaces of a vector space $V$. Then $\dim(W_1 \cap W_2)=\dim(W_1)$ if and only if $W_1 ...
2
votes
4answers
177 views

Orthogonal matrices proof

Let $\upsilon _{n} $ be the set of all $n \times n$ orthogonal matrices(for all fixed $n$). Show that $\upsilon _{n} $ is not a subspace of $\ M _{n\times n} $. Thank you ! Additional: Suppose $A ...
0
votes
2answers
77 views

Are the lengths from this recursive construction a geometric sequence?

In his 1999 review of Edward Tufte's Visual Explanations in the Notices of the AMS (third page), Bill Casselman gives a very pretty proof of the irrationality of the golden mean. More precisely, ...
1
vote
1answer
65 views

Prove that the Iwata function is Submodular

The Submodularity property for $f: 2^V \rightarrow \mathbb{R}$ is defined as: $f(X) + f(Y) \geq f(X \cup Y) + f(X \cap Y)$ where $X, Y \subseteq V$ While the Iwata function is defined as: ...
2
votes
2answers
142 views

Finding the solution to this specific recurrence relation

What would be the solution to $a_n = 7a_{n−2} + 6a_{n−3}$ with $a_0 = 9$, $a_1 = 10$, and $a_2 = 32$ I can find it for a specific value of (n), but not for just a general solution. Thanks!
3
votes
0answers
35 views

fancy about some properties of kernel functions at infinity

Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not: $1.$ If $K(x,t)$ is bounded ...
1
vote
1answer
63 views

Proof Technique and Factorials

I need to prove that $\;n!+m$ is divisible by $m$ for all integers $n \ge 2$ and $1 \le m \le n$.
1
vote
0answers
99 views

Proving a specific recurrence relation theorem

I'm trying to come up with a proof for this theorem: Let $c_1$ and $c_2$ be real numbers with $c_2 != 0$. Suppose that $r^2 - c_1 r - c_2 = 0$ has only one root, $r_0$. A sequence $\{a_n\}$ is a ...
1
vote
2answers
558 views

Finding a solution to a recurrence relation

Find the solution to $$a_n = 5a_{n−2} − 4a_{n−4}$$ with $$a_0 = 3$$ $$a_1 = 2$$ $$a_2 = 6$$ $$a_3 = 8$$ My answer: Observe that the degree of recurrence is 4. Hence, the characteristic equation is: ...
3
votes
1answer
244 views

Finding a Linear Recurrence Relation

A model for the number of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two previous years. ...
1
vote
0answers
92 views

Is this theorem proof correct?

I'm trying to prove this theorem: Let $c_1$ and $c_2$ be real numbers with $c_2 \ne 0$. Suppose that $r^2 − c_1r − c_2 = 0$ has only one root $r_0$. A sequence $\{a_n\}$ is a solution of the ...
1
vote
2answers
117 views

Strategies to prove inequalities with interval notation

How to prove a inequalities with interval notation, for example: Find minimum of $a^3+b^3+c^3$ with $a,b,c \in [-1;\infty), a^2+b^2+c^2=9$
0
votes
2answers
50 views

Help with real number proof strategy

Let $x, y, z \in \mathbb R$. Prove that $\displaystyle \frac{|x-y|}{1+|x-y|} \leq \frac{|x-z|}{1+|x-z|} + \frac{|z-y|}{1+|z-y|} \; $ Help on how to start with this one? Real lost.