The proof-strategy tag has no wiki summary.
2
votes
2answers
30 views
Any practical difference between the metrics $d_1=\sup\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ and $d_2=\max\{\left|x_j-y_j\right|:j=1,2,…k\}$?
I've been asked to do a proof showing that $d_1\left({x,y}\right)$ is a metric on $\mathbb{R}^k$, but is there any difference between this and $d_2\left({x,y}\right)$, for which I've already done a ...
3
votes
1answer
65 views
$V=W_1\oplus\cdots\oplus W_k \iff \dim(V)=\sum{\dim(W_i)}$
If $W_1,\dots, W_k$ are subspaces of a finite dimensional vector space $V$ such that $W_1+\cdots+W_k=V$, and I want to show that $V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{W_i}$, then ...
0
votes
0answers
37 views
property of an increasing or decreasing function
For $x \in \mathbb{R}$, and $f(x)$ an increasing function, can we prove whether
$$ af(x)\lesseqgtr f(ax) $$
for $a >0$? If we have additional information that $f$ is homogeneous of some degree, ...
2
votes
3answers
90 views
Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?
I want to use Cantor's diagonalisation argument to prove that the set S of surjective functions of the form $\Bbb{N} \to \Bbb{N}$ is uncountable. The normal procedure is creating a matrix and filling ...
2
votes
0answers
48 views
Proving identity involving sum
I'm stuck trying to prove the following identity, which is seemingly correct (from mathematica):
$$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose ...
3
votes
2answers
86 views
REVISTED$^1$: Circumstantial Proof: $P\implies Q \overset{?}{\implies} Q\implies P$
To prove that if a matrix $A\in M_{n\times n} ( F )$ has $n$ distinct eigenvalues, then $A$ is diagonalizable is enough to show that the opposite holds? That is, if $A$ is diagonalizable, then $A$ has ...
1
vote
1answer
46 views
Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.
I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
3
votes
3answers
39 views
Prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ is a metric on $\mathbb{R}^k$
I am trying to prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,...,k\}$ is a metric on $\mathbb{R}^k$. So far I know that $\text{dist}\left({x,y}\right) \leq ...
4
votes
4answers
85 views
Proving one function is greater than another
How can I prove $f(x)$ $>$ $g(x)$ for all $x > 0$ given $f(x) = (x+1)^{2}$ and $g(x) = 4qx$ where $q$ is a constant in $(0, 1)$?
My approach was to show that $(x+1)^2 > 4qx$ for the interval ...
0
votes
2answers
48 views
If $E = \{ x \in \mathbb{R}: \sin(\frac1{x}) = 1\}$ then $l = 0$ is a limit point of E
If $E = \{ x \in \mathbb{R}: \sin\left(\frac{1}{x}\right) = 1\}$, then $l = 0$ is a limit point of $E$.
I have a proof here but I don't quite understand a few points, I hope someone can explain it a ...
-6
votes
2answers
59 views
Help with Theorem III.3.11 in Hungerford's algebra book
I need help to prove part (i) of this theorem which I couldn't prove.
Any help would be appreciated. Thanks in advance.
-2
votes
0answers
67 views
Best mathematical proofs [closed]
For you wich are the best mathematical proofs?
I can remember Furstenberg´s proof of the infinitude of primes that really amaze me.
I am very interested in this kind of proof that really can ...
2
votes
2answers
48 views
Suppose $f$ is a real-differentiable function on $[a,b]$ and suppose $f'(a)<c<f'(b)$. Prove then there is a point $x \in (a,b)$ such that $f'(x)=c$
This is what i have:
Put $g(t) = f(t) - ct$.
Then $g'(a)<0$ so that $g(t_{1}) < g(a)$ for some $t_{1} \in (a,b)$ and
$g'(b)>0$ so that $g(t_{2}) < g(b)$ for some $t_{2} \in (a,b)$.
...
2
votes
1answer
16 views
Let $f$ be defined on $[a,b]$, Prove that if f has a local maximum at a point $x \in (a,b)$, and if $f'(x)$ exists, then $f'(x)=0$
Is this proof correct:
Let's choose a $\delta$ to that
$a < x - \delta < x < x + \delta < b$
If $ x - \delta < t < x$
then $\frac {f(t) - f(x)} {t-x} \geq 0$
Letting $t ...
4
votes
0answers
58 views
Good examples of proofs in mathematics exemplary of creative reasoning [closed]
Just what the title says. I'm not looking for any proofs that require specialized knowledge past the very fundamentals of real analysis. I'm looking for proofs for important results (don't have to be ...
1
vote
1answer
55 views
Prove (without quoting any theorems) that polynomials on [0,1] are continous
I'm confused as to go about this problem. I feel as if we have to show that
$P [0,1] \in C^{0}[0,1]$
by letting
$f = a_{n}x^{n} + a_{n-1}x^{n-1} + .... + a_{1}x^{1} + a_{0}$
We must show that ...
0
votes
1answer
36 views
Bilinear Forms: An Initial Condition Proof
Let $B$ be a bilinear form on a finite dimensional vector space $V$. Suppose that for any nonzero vector $v \in V$ there exists a $w \in V$ such that $B(v, w)\neq 0$. Prove that for any linear ...
0
votes
2answers
35 views
Odditiy: An Analysis of Skew-Symmetric $n\times n$ Matrices
Let $A \in M_{n×n}(\mathbb{R})$ be a skew-symmetric matrix, i.e., $A^t = −A$. Prove that if $n$ is odd, then $\det{A} = 0$.
5
votes
1answer
38 views
$\inf A = -\sup(-A)$
Let $A$ be a nonempty subset of real numbers which is bounded below. Let $-A$ be the set of of all numbers $-x$, where $x$ is in $A$. Prove that $\inf A = -\sup(-A)$
So far this is what i have
...
1
vote
1answer
38 views
Prove that a polygon with nonnegative area is determined by at least three points.
How do you prove this statement in geometry?
A polygon with nonnegative area can't be formed with fewer than 3 points.
1
vote
0answers
37 views
Using Compactness to obtain an inequality.
I'm reading a proof that the Hausdorff dimension of the Cantor set is $\frac{\log 2}{\log 3}$ using the definition of Hausdorff dimension.
The lower bound of the proof requires a lemma.
(the problem ...
3
votes
1answer
62 views
Proof the following trig series
Prove that
$$\frac{ \sin x}{ \cos x}+\frac{\sin2x}{\cos^{2}x}+\frac{\sin3x}{\cos^{3}x}+\cdots+\frac{\sin nx}{\cos^{n}x}=\cot x-\frac{\cos(n+1)x}{\sin x \cos^{n}x}$$
I am not necessarily looking for a ...
4
votes
2answers
146 views
Real Numbers is a subset of Complex Numbers?
So, I was taught that $\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$
But, since the complex numbers' definition is $\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}$,
doesn't that mean ...
1
vote
1answer
46 views
Combinatorics identity sum of
Prove that:
$$\sum^{n}_{k=0}\binom{k}{2n-k}2^k = 2^{2n}$$
By using only combinatorics identities.
0
votes
1answer
30 views
prove $f^{-1}(B)=A$
I am given $A_1$, $A_2 \subseteq A$ and $B_1$,$B_2 \subseteq B$. and the function $f: A \rightarrow B$
I want to prove that $f^{-1}(B)=A$.
I just assume that here one is talking about ...
3
votes
2answers
119 views
Where is wrong in this proof [duplicate]
Suppose $a=b$.
Multiplying by $a$ on both sides gives $a^2 = ab$. Then we subtract $b^2$ on both sides, and get
$a^2-b^2 = ab-b^2$.
Obviously, $(a-b)(a+b) = b(a-b)$, so dividing by $a - b$, we find
...
2
votes
1answer
22 views
Residue of a 1-form in a Riemann Surface does not depend of the chart
Let's suppose that $X$ is a Riemann Surface, $\omega$ is a meromorphic 1-form in $X$ and let $p$ be a pole of $\omega$ of order $M$. I want to show that the residue of $\omega$ at $p$, defined by
$$
...
0
votes
1answer
14 views
Clues to prove average in T is minor or equal than average in a smaller inner interval.
Suppose I want to prove (or disprove) this assertion
Let $f$ be a discrete function, $T,h,k$ are constants
So these terms are averages over $T$ and over $h$
$\sum\limits_{i=0}^{T}\frac {f(i)}{T}$ ...
7
votes
2answers
166 views
Prove $\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \, dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \, dx$
Prove that:
$(1)$$$\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \ dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \ dx$$
$(2)$$$\int_0^{\infty } \frac{1}{\sqrt{8 x^3+x+7}} \ dx>1$$
What I do for ...
0
votes
0answers
41 views
Proof contraction differentiable function
$g$ : $R$ $\rightarrow$ $R$ be a diferentiable function such that $-1$ < $a$ < $b$ < $0$ where for $y$ $\in$ $\Re$,
$a$ $\le$ $g'(t)$ $\le$ $b $
Prove that $g(t) = t + f(t)$ is a contraction ...
1
vote
0answers
34 views
Proving that the circumcenter is the centroid
Given a triangle and its centroid, we know that the 3 line segments between the centroid and each of the vertices of the triangle divide the triangle into three smaller triangles. Prove that the ...
1
vote
2answers
61 views
Help with proofs: Show that $AA^T$ and $A^TA$ are symmetric
I need help with a proof for my liner algebra class.
If $A$ is a square matrix, then $AA^T$ and $A^TA$ are symmetric.
I have no idea where to start!
3
votes
1answer
34 views
$\mathbb{Z}/m\mathbb{Z}$: A Complete Set of Representatives
So, I'm letting ${\scr{A}}=\{a_1,\dots,a_n\}$ be a complete set of representatives (C.S.R.) for $\mathbb{Z}/m\mathbb{Z}$. I'm considering all $b\in \mathbb{Z}$ and seeing if ${\scr{B}}=\{a_1+b, a_2+b, ...
5
votes
3answers
111 views
Prove that $(a+1)(a+2)…(a+b)$ is divisible by $b!$ [duplicate]
The problem is following, prove that:
$$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$
I've tried solving this problem using mathematical induction, but I ...
2
votes
2answers
36 views
proof by induction to demonstrate all even Fibonacci numbers have indices divisible by 3
I am practicing proof by induction, and would like to use induction to prove the following hypothesis about the Fibonacci numbers:
$$(\forall n\ge0) \space 0\equiv n\space mod \space 3 \iff 0 \equiv ...
1
vote
1answer
32 views
$\operatorname{rank}(A\in M_{m\times n}(F)) =m \implies \exists~B\in M_{n\times m}(F)$ s.t. $AB=I_m$
Let $A ∈ M_{m×n}(F)$ be a matrix with $\operatorname{rank}(A) = m$. I just need some help showing that there exists a matrix $B ∈ M_{n×m}(F)$ such that $AB = I_m$.
2
votes
2answers
57 views
How do I prove the arithmetic-geometric mean inequality?
I am following along with this bare-bones proof of the arithmetic-geometric mean inequality with two real numbers. I'm having difficulty understanding the logic behind this step:
$$
...
3
votes
1answer
39 views
Rice's theorem_Theory of computation
Is there any body tell me, where is wrong in this proof
Problem: The set of number of turing machine that has 5 state is decidable or not?
Answer: The set is obviously 'Set of partial computable ...
4
votes
3answers
73 views
Extreme Value Theorem Proof (SPIVAK)
Them: If $f$ is continuous on $[a,b]$, then there is a $y$ in $[a,b]$ such that $f(y) \geq f(x)$ for each $x \in [a,b]$
OKay first of all how on earth does one come up with $g(x)$? It just ...
1
vote
1answer
51 views
Is this proof on the product of $X$ OK?
Let $X^2$ be star $\sigma$-compact and $F$ be a closed subset in
$X^2$. If $\mathcal{U}$ is an open cover of $F$, then there exists a
$\sigma$-compact subset $A$ of $X$, such that $F \subseteq
...
1
vote
1answer
41 views
A modified Buffon's needle
A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it.
What is the probability that the needle lies between the two lines?
I can see how ...
2
votes
1answer
47 views
Are there examples of theorems proved via proper (i.e. non-conservative) extensions?
This is not a question about set theory specifically, but lets talk about ZFC just for concreteness
Suppose we have a sentence $\phi$ in the language of ZFC, and a proof that either $(\mathrm{ZFC} ...
0
votes
1answer
37 views
Finding a reccurence relation for the following problem
A circular disk is cut into n distint sectors, each shaped liek a piece of pie and all meeting at the center point of the disk. Each sector is to be painted red, green, yellow, or blue in such a way ...
-2
votes
1answer
51 views
Given the following recurrence relation, prove using mathematical induction
How can we prove this using mathematical induction?
$m_1 = 0$
$m_k = m_{\lfloor (k/2) \rfloor} + m_{\lceil (k/2) \rceil} + k-1$ for all integers $k \geq 1$
Prove using mathematical induction that ...
0
votes
6answers
110 views
Finding the number of subsets of S
How can we find the number of subsets of $S=\{1,2,3,...,10\}$ that contain neither 5 nor 6?
Thanks!
0
votes
2answers
49 views
Use the binomial theorem to expand
How can we expand this using the binomial theorem?
$(x^2 + 1/x)^7$
4
votes
1answer
63 views
How does one DERIVE the formula for the maximum of two numbers
I want to derive (not prove that this is true) the formula
$\max (x,y) = \dfrac{x + y + |y-x|}{2}$
I was reading a proof (which they have the result ahead of time already) that we do cases and then ...
1
vote
2answers
94 views
A typo in Spivak's solution?
Problem
Solution
I honestly cannot figure out what he is doing. On one hand, I think Spivak wants to write $|\phi(b)/b^n| > 1/2$ instead of $|\phi(b)/b^2| < 1/2$. On the other ...
1
vote
2answers
43 views
Induction on the Fibonacci sequence?
Prove by induction that the $i$th Fibonacci number satisfies the equality:
$$F_i = \frac {\phi^i - \hat\phi{}^i}{\sqrt5}$$
where $\phi$ is the golden ratio and $\hat\phi$ is its conjugate.
...
3
votes
2answers
42 views
Show that $\exists A \subset \mathbb{R}$ such that $\forall x$ $\in \mathbb{R}$, we may write $x$ uniquely as $x=a+q$, where $a\in A,q\in\mathbb{Q}$.
Not sure where to go with this one. Clearly will have to use the axiom of choice at some point. I haven't been able to think of a good example for the set A. Once we've got that, it'd be a matter of ...
