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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1answer
22 views

Trouble Understanding Proof Of Invariant Relationship

In part of a proof I am reading this is stated: $2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ) + (a_n + c_n )^2 + (b_n + d_n )^2 ≥ 2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ).$ (1) From this invariant inequality ...
2
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3answers
49 views

Injective Ring Homomorphism

I seem to be having the wrong impression of what $p$ stands for; is $p(x)=x(x+1)(x+2)$ or is it something else? Clarification would be appreciated so that I can complete the lemma below. Consider ...
3
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1answer
43 views

Proving $P$ by proving $\neg Q$ and knowing $P\lor Q$

This may sound silly. I used to remember studying this in physics class and I thought of asking it in physics.stackexchange and then later I decided to ask it here itself. Let's say, under some ...
0
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0answers
28 views

How do I show $G_0$ and $G_1$ are conjugate subgroups? Please improve my answer.

Is my solution below correct? Please read through it and tell me if it seems complete or to make sense. Question: Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the ...
0
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2answers
34 views

How to find the limit of the sequence

$X_0 := 2$ and for $X_n$: $X_{n+1} = \frac12 X_n + \frac1X_n$ I know that the sequence is monotone decreasing and I am not sure how to find its limit maybe with Cauchy and partial sums but still I ...
0
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1answer
35 views

If $f$ s twice differentiable and satisfies the following constraints, prove $f'(0)>-\sqrt 2$

Let $f$ be a twice differentiable function on the open interval $(-1,1) $such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0 $and $f''(x) \le f(x)$, for all $ x\ge 0$. Show that ...
2
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0answers
49 views

A more detailed, rigorous proof that a suspension space is not necessarily contractible

Is my answer/proof correct? Please help me make my proof more rigorous and detailed. I need everything to be absolutely clear. Question: Let $X$ be a topological space. The suspension of $X$, ...
6
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1answer
61 views

Proving the Cone is Contractible: Is my Proof correct?

Is my answer/proof correct? Please help me make my proof more rigorous and accurate. I need everything to be absolutely clear and rigorous. Thank you. Question: Let $X$ be a topological space. The ...
1
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3answers
28 views

How to show that a complex function have a branch in a domain

I've given as homework to show that the function $$f(z)=\sqrt{\frac{z+1}{z-1}} $$ has a branch on $G = \mathbb C \backslash [-1,1] $. I'm having a hard time in finding the way to approach this kind ...
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4answers
49 views

Prove that lim of x/(x+1) = 1 as x approaches infinity

I want to prove that $$\lim\limits_{x\to \infty} \frac{x}{x+1}=1$$ I know that I need to show that: $$\left|\frac{1}{x+1}\right| \lt \epsilon$$ But I'm not sure how to manipulate it. Any help or hint ...
1
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0answers
35 views

Prove that the continuous $f: \mathbb C \to \mathbb R$ has a global max and min

I am having this continuous transformation $f: \mathbb C \to \mathbb R$ and $\ f\ (\mathbb C)$ is bounded Now I have to prove that there are a global maximum and a global minimum. My thoughts: I ...
6
votes
0answers
61 views

Examples of useful, insightful and interesting hand-waving

I am really amused by the answers to this question on "Most harmful heuristic" posed on MathOverflow, from which I've benefited a lot. However, it seems to me that some hand-waving may be really ...
6
votes
2answers
103 views

How to explain to a layman why Fermat's Last Theorem involves non-trivial math?

Fermat's Last Theorem states, given$$x^n + y^n = z^n$$ no three integers $x,y,z$ will satisfy the equation given integer value of $n$ greater than two. On the surface this seems like something that ...
4
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6answers
398 views

Question on induction technique

When one uses induction (say on $n$) to prove something, does it mean the proof holds for all finite values of $n$ or does it always hold when even $n$ takes $\pm\infty$?
2
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5answers
108 views

Is there a proof that $\int \frac {dx}{x}=\ln |x|+c$?

Is there a proof that $$\int \frac {dx}{x}= \ln|x|+c$$ for $x\neq 0$ I would be interest for any replies or any comment.
0
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1answer
19 views

Proof if $n_k < n_{k+1}$ for all $k \in \mathbb{N}$, then $n_k \geq k$ for all $k \in \mathbb{N}$.

So if we proceed by induction on $k$, the base case $k = 1$ works since $n_1 \geq 1$ is true because $1$ is the smallest integer in $\mathbb{N}$. For the induction hypothesis, we have that $n_k \geq ...
0
votes
1answer
15 views

Deduce an inequality by using Bernoulli's Inequality

Deduce $c^n\geq c \forall n\mathbb\in{N},c>1$ What I have tried is $$\text{Let }x=c-1$$ Then I substitute it into the Bernoulli's inequality, that is $$c^n\geq1+n(c-1)\geq 1+nc-n\geq nc+1$$ How ...
0
votes
3answers
17 views

Proving a surjective function by given property

Suppose $f:E\rightarrow F$ and for any $A\subset F,A=f(f^{-1}(A))$. Show that f is surjective. What i have tried is $$\text{Let }y\in A $$ $$\{y\}\subset A$$ $$\{y\}= f(f^{-1}{\{y\})}$$ And i stuck ...
0
votes
4answers
32 views

Counting candies in boxes

There are $5$ boxes containing $80$ candies. After taking $\frac{1}{5}$ of the candies in the first box and putting them in the seconf one, we take $\frac{1}{5}$ of the candies in the second box and ...
0
votes
0answers
51 views

Don't understand proof that if $\{x_n\}$ is Cauchy and if some $x_{n_k} \rightarrow x$, then $x_n \rightarrow x$

So by definition of Cauchy, for all $\epsilon > 0$ and $i, j \in \mathbb{N}$, there exists an $M$ such that for all $i, j \geq M$, then $|x_i - x_j| < \epsilon'/2$ if we let $\epsilon = ...
0
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1answer
38 views

Don't understand proof that convergence implies Cauchy

So we are given that $x_n \rightarrow x$, so we can let $\epsilon = \epsilon'/2$ and there definitely exists an $N$ such that for all $n \geq N$, $|x_n - x| < \epsilon'/2$. Also by the triangle ...
2
votes
2answers
45 views

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective.

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective. This is rough. I've been staring at this one for a while now. I get stuck on the ...
0
votes
1answer
42 views

Beginning Haskell - cannot understand proof

I've just started reading "Thinking Functionally with Haskell" by Richard Bird In the preface he states : And after stating the proof he also states the proof will be used throughout the book. ...
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1answer
42 views

Mathematical induction to proof [on hold]

Prove that $$\frac{1}{1}+\frac{1}{4}+\frac{1}{9}+...+\frac{1}{n^2} = \sum_{k=1}^n \frac{1}{n^2} \leq 2-\frac{1}{n}$$ Why would the answer said that 'the summation of (n+1) term from k 1/k^2 ...
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2answers
20 views

If $\sup A = 5$ and $B = \left\{ 3a \mid a \in A \right\}$ then $\sup B = 15$

Prove that if $A \subset \mathbb{R}$, $\sup A = 5$, and $B = \left\{ 3a \mid a \in A \right\}$, then $\sup = 15$. I tried to do contradiction by assuming the hypothesis and that there is a number ...
2
votes
1answer
51 views

Ways of proving that $A=0$

I was solving a problem where you had to prove that some number $=0$. My strategy was to show that $Ak=A$ for some $k$ not equal to 1, hence $A(k-1)=0$ from which it follows that $A=0$. Abstracting ...
0
votes
2answers
59 views

Differentiability of a function

How can I prove that the function $x^{n+\frac{1}{2}}$ is differentiable? Do I split it up into $x^n$ and $x^{\frac{1}{2}}$? Any suggestions would be great. Thanks
0
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1answer
25 views

Continuity on Integrals

Suppose that f(x)>= 0 for all x in [a,b] and f is continuous at x0 in [a,b] and f(x0) > 0 Prove that the integral from a to b of f is greater than zero. Can i prove this using the bounded theorem ...
-3
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2answers
39 views

Let A be a nonempty set, and define function $f\colon A\to P(A)$ by $f(a)=\{a\}$. Show that $f$ is one-to-one but not onto. [on hold]

Please help on a homework problem. How can I show something is one-to-one BUT not onto?
1
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1answer
16 views

show that a loopless graph $G$ contains a bipartite spanning subgraph $H$ such that $d_H(v) \ge \frac{1}{2} d_G(v)$ for all v $\in$ V.

The hint in the appendix of book says that bipartite subgraph with with largest possible number of edges has such a property, but I don't know how to use this hint! any help would be appreciated.
0
votes
1answer
67 views

Prove that limit goes to inf

Let $f:\mathbb R \to \mathbb R$ be such that $f(x), f'(x)$ and $f''(x)$ are all positive for each $x \in \mathbb R$. Apply the MVT to $f$ on each interval $[n,n+1]$ for $n=1,2, 3,\dots$ and show that ...
0
votes
1answer
20 views

Can anyone explain how we use the linear extension theorem?

I am having problems using the linear extension theorem. For example: Let V be finite-dimensional, and let W ⊂ V be a proper subspace of V . Fix a vector v0 ∈ V such that v0 is not in W. Show that ...
0
votes
2answers
69 views

How to prove that $\lim_{x \to \infty} x = \infty$

Please refrain from using logic symbols, as I do not understand those. So, this is the question: $$\lim_{x \to \infty} x = \infty$$ Proving this using the actual formal definition of a limit. So ...
2
votes
3answers
37 views

Proving if $A \subseteq B$ and $A \nsubseteq C$ then $B \nsubseteq C$

This is one of the problem I have been solving in Velleman's How to prove book: Prove that if $A \subseteq B$ and $A \nsubseteq C$ then $B \nsubseteq C$ This is my solution: Suppose $A ...
2
votes
1answer
20 views

Point in a rectangle

$ABCD$ is a rectangle and $P$ is a point in the same plane. If the perpendicular through $C$ to $AP$ and the perpendicular through $B$ to $DP$ intersect at $Q$, prove that $PQ \parallel AD$. ...
3
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1answer
31 views

Show g is unbounded above if g and g' are increasing

Suppose $g$ is a function defined on the set of real numbers where $g(y)$, $g'(y)$, and $g''(y)$ are all greater than $0$ for all $y \in \mathbb R$. Show that $g$ is unbounded above as $y$ approaches ...
2
votes
2answers
17 views

Proving that median of list $[x_1,x_2,…,x_n]$ minimises the sum $\sum_{i=1}^{i=n} |x_i-m|$ where $m$ is some number

The problem is in the title. Here is a detailed description: Let's say we have list $[x_i]_{i=1}^{i=n}$ where $x_i\in\Bbb{N}$. I want to pick such $m\in\Bbb{N}$ which minimises the sum ...
1
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4answers
63 views

Integrals are equal

Suppose that $f$ is integrable on $[a, b]$. Prove that there is a number $x \in [a, b]$ such that $$\int_a^x f(t)\,dt = \int_x^b f(t)\,dt .$$ Show by example that it is not always possible to choose ...
3
votes
2answers
112 views

Rolle's Theorem with roots

Let $f : [a, b] \to \mathbb R$ be $n$ times differentiable and have $n+1$ distinct roots (i.e. solutions of $f(x) = 0$) in $[a,b]$. Show that there is an $x \in [a, b]$ such that the $n^{\text{th}}$ ...
0
votes
1answer
12 views

Infinite sequence of real numbers converging to x and y

So the question is: Suppose $x_i$ and $y_i$ are infinite sequences of real numbers converging to x and y. Show that $(x_i + y_i)$ converges to $x+y$. Show that $x_iy_i$ converges to $xy$. Here's ...
3
votes
2answers
44 views

Inverse using Fundamental Theorem of Calc

Find $(f^{-1})'(0)$ if $f(x) = \int_1^x{ \cos(\cos t)dt}$ So question about this. For the problem there was no interval given so that the function $\cos(\cos t)$ was strictly increasing (which we ...
0
votes
1answer
15 views

How to find the characteristic polynomial of this transformation?

Let V be a finite-dimensional inner product space, and let W ⊂ V be a subspace. Let T : V → V be the linear transformation “orthogonal projection onto W”: T(x) = ProjW x. Show that T is ...
2
votes
2answers
43 views

$X:\Omega \to \mathbb{N}$ is random variable, How to prove that $E[x]=\sum_{i} \Pr(X\ge i)$?

I'm stuck with this proof: $X:\Omega \to \mathbb{N}$ is random variable, prove that $\mathbb{E}[X]=\sum_{i=1,2,3...} \mathbb{P}(X\ge i)$? How I'm proving it? I'm starting with the definition: ...
1
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1answer
30 views

Vector question involving an operator!

So, here's the problem: An operator H capable of operating on vector x, is defined in terms of a given vector a by: H x=(a * x) where $*$ representes vector product Given that ...
0
votes
2answers
36 views

Sum of odd numbers is odd if each of the natural numbers is odd

The question is: Proof that the sum of an odd number of natural numbers is odd if each of the natural numbers is odd. Here's what i tried already but it didn't work: $\sum_{i=0}^n i = 2n-1$ but ...
0
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1answer
29 views

interval proof using points

So this is for my advanced calculus class (Real Analysis II) which is a proof class. The question is: If $a<b$ are points in an interval $D$, show that $[a,b]$ $\subset$ $D$. I feel like its ...
0
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2answers
72 views

Use the Mean Value Theorem to show that if $|f'(x)| ≤ C<1$, then $f$ has at most one fixed point

Use the Mean Value Theorem to show that: if $|f'(x)| ≤ C < 1$ $\forall x$, then $f(x) = x$ has at most one solution. So using the Mean Value Theorem I know that $$-1<-C\leq ...
1
vote
3answers
96 views

Continuous functions and infinum

Let $f:\mathbb R \to \mathbb R$ with $f(-2)=4$ and $f(3)=7$. Let $S:=\{x \in [-2,3]\mid f(x)\geq 5\}$. Then $\alpha:=\inf S$ exists. If $f$ is continuous at $\alpha$, show that: (a) ...
0
votes
2answers
33 views

I didn't figure out how the result in part (i) can help in (ii). Anyone has any idea??

The determinant turns out to be -3 in part (i) How can this help in showing that the 4 vectors in the end are linearly independent?
0
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2answers
26 views

The product of two nonnegative, improperly integrable functions is also improperly integrable.

True or False: The product of two nonnegative, improperly integrable functions is also improperly integrable. I was given both the problem and the proof that may or may not be true. I think the ...