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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1
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1answer
12 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 ...
-3
votes
4answers
46 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
vote
0answers
32 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
56 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
99 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
votes
6answers
387 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
votes
5answers
106 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
votes
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
31 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
votes
1answer
37 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
41 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. ...
-8
votes
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 ...
1
vote
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
votes
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
votes
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
vote
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
66 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
65 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
votes
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
vote
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
vote
1answer
29 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
votes
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
votes
2answers
71 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
95 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
votes
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 ...
4
votes
2answers
41 views

If $f$ is continuous on $[a,b)$ and $[b,c]$, then $f$ is Riemann integrable on $[a,c]$.

True or False: If $f$ is continuous on $[a, b)$ and on $[b, c]$, then $f$ is Riemann integrable on $[a, c]$. I was unsure if the $)$ in $[a,b)$ completely changed the problem and made it false and I ...
3
votes
3answers
61 views

Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism

I'm working on the following problem: Let $G$ be a group. Prove that $G$ is abelian if and only if $\varphi(g) = g^2$ is a homomorphism. My solution: First assume that $G$ is an abelian group ...
6
votes
2answers
51 views

How do I close the gap between intuitively knowing something is true vs being able to prove it?

For example, one of my review problems is: Let $S_k$ be the kernel of $T^k$. Show there is a $K$ such that $S_K = S_{K+1} = \cdots$ Somewhere in the back of my brain there's an intuition that told ...
1
vote
1answer
77 views

How learn proofs? [closed]

I'm in high school, and I'd like to know how you guys learn proofs? What method and attitude you guys take when learning proofs? For example, when learning the proof of something simple like the sum ...
1
vote
2answers
22 views

Prove proposition on real numbers and uniqueness.

How would I go about proving the following proposition. Do I have to prove uniqueness, or that if $x^2 = r$, then $x = \sqrt r$? Prove given any $r \in \mathbb R\gt 0$, the number $\sqrt r$ is ...
1
vote
0answers
13 views

Proof of Polyates Lemma

In Sbiis Saibian's site I came across Polyates Lemma which states that $$(b \uparrow^k m) \uparrow^k n\ <\ b\uparrow^k (m+n)$$ for all positive integers b,m,n,k with $b\ge 2$ and $k\ge 2$. He ...
0
votes
1answer
13 views

Approximating a field by perfect fields.

Let's consider an arbitrary field $K$ and raise the following question: in which sense can we approximate $K$ by a perfect field? Any reasonable notion of approximation by a perfect field should admit ...
0
votes
3answers
44 views

Help explain the end of this proof for infinitely many primes?

by contradiction, assume finitely many primes $p_1, p_2,\cdots, p_k$. let $N = p_1p_2\cdots p_k + 1$. Note $N > 1$. Now, by the fundamental theorem of arithmetic, there exists a number $p_j$, where ...