2
votes
1answer
70 views

Proving a set of numbers has arithmetic progressions of arbitrary length, but none infinite

For each real number $x$, let $[x]$ be the largest integer less than or equal to $x$. For example, $$[5] = 5$$ $$[7.9] = 7,$$ and $$[−2.4] = −3.$$ An arithmetic progression of length $k$ is a ...
1
vote
1answer
113 views

Understanding last step of a proof that “two trajectories cannot cross at a finite value of t” (Phase trajectories/nodes)

Note: This proof prefaced critical points at the origin for coupled first order ODEs. It was done before showing the asymptotically stable and unstable critical points: Improper, Proper, Spiral, ...
1
vote
0answers
40 views

Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in ...
1
vote
3answers
29 views

Where does my proof of uniform continuity fail?

I am trying to prove that $f:R \to R f(x)=\sin x$ is uniformly continuous. I have said: Fix $\epsilon > 0$ and $\delta=\epsilon$ $|\sin x - \sin y| \le |\sin x| - |\sin y| \le 1 - 1 = 0 ...
0
votes
1answer
33 views

Proving a function is not uniformly continuous.

I am using the definition: $(∃ε > 0)(∀n ∈ N)(∃ x_n, y_n ∈ (0,1])[(|x_n − y_n| < δ_n =1/n) ∧ (|f(x_n) − f(y_n)| ≥ ε)]$ to prove that $1/x^2$ is not uniformly continuous. In the solution I am ...
0
votes
0answers
28 views

Proving a Cauchy Sequence without an explicit formula for $(a_n)$ [duplicate]

How would I prove that for all $n$ in the natural numbers, $|a_{n+1} - a_n| \le 1/2^n$ $(a_n)$ is a Cauchy Sequence? I have tried to use the standard definition and the triangle inequality, but ...
0
votes
2answers
37 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
1
vote
1answer
14 views

Limit of a function proof verification

My proof: By Bernoulli Equation $(a^n+b^n)^{1/n}=b(1+(na)/b)^{1/n}$ By definition of a limit, fix $\epsilon > 0$ and $N>(b\epsilon^n)/a$ Then, $|a_n - b | = ...
1
vote
2answers
24 views

Series, limits and convergence.

Theorem $\,\bf3.3.1.\;$ If the series $$\sum_{n=1}^\infty a_n$$ is convergent then $\lim\limits_{n\to\infty}a_n=0$. Proof. Let $s_n=\sum_{k=1}^n a_k.$ Then by the definition the limit $\lim_n ...
2
votes
3answers
73 views

Proving analytical statement, Intermediate Value Theorem

Let's define $f$ as a continuous function with $f:[0;2] \to \mathbb{R}$ and $f(0) = f(2)$. Now, I want to show that: $$\exists x_0 \in [0;1]:f(x_0) = f(x_0 + 1)$$ I tried to plot a few functions in ...
6
votes
2answers
222 views

Proof of continuity - (ε-δ) definition - Can anyone check this?

I've been trying to get my head around this problem for quite some time by now. I want to prove that $$f(x) := \left|\frac{x-1}{x^2+1}\right|$$ is continuous for $$x_0 = -1$$ Now, in order to prove ...
0
votes
1answer
21 views

Trying to show that a vector value function has equal mixed partial derivatives

Let $f: R^n \to R^n$. $||x||$ is Euclidean norm. Define $f(x) = xg(||x||)$. where $g: [0, \infty) \to R^n$ is differentiable on $(0, \infty)$. $g$ is not constant. I want to show that every $i \neq ...
1
vote
2answers
52 views

Prove that f(x) is uniformly continuous

Show that $f(x)=x^2cos(\frac{1}{x^2})$ on (0,1). Let $\epsilon > 0$ and $x,y \epsilon (0,1)$. $| f(y) - f(x)| = |y^2cos(\frac{1}{y^2}) - x^2cos(\frac{1}{x^2})|$. After this I do not know how to ...
0
votes
2answers
33 views

How to quantify this statement

How do I quantify this statement? Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. When I am trying to prove this, I am led (in the course of ...
1
vote
1answer
87 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
0
votes
1answer
45 views

General conceptual confusion relating to vacuous proofs and quantifier help

I need to prove the statement: Let $x \in \mathbb{R}$. Prove that $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. So I start with the forward implication: If $1 ≤ ...
0
votes
2answers
49 views

Analysis: Prove the converse

It can be shown that if $\lim_{n\to\infty} a_n = L$, then $\lim_{n\to\infty} |a_n| = |L|$. Is the converse of this result true?
2
votes
4answers
77 views

A question about metrizability

In a lecture in Topology I had earlier this week, I was told (without proof) that not every topological space $(X,O)$ is metrizable, i.e, it is impossible to find some metric $d$ such that $O$ and ...
1
vote
2answers
92 views

$\epsilon - N$ definition of a limit of sequence problem

i have a question i cannot seem to solve! i would really appreciate help if possible. please explain how to solve this question from textbook, i really want to learn but i cant $$\lim \limits_{n \to ...
0
votes
2answers
18 views

Show that for any $s \in S$ where $S = $ {$s \in \mathbb{N} | \exists m,n \in \mathbb{Z}$, we have $d|s$ where $d$ is the smallest element

Explain why the set $S = $ {$s \in \mathbb{N} | \exists m,n \in \mathbb{Z}, s = ma+ nb $} has a smallest element, call it d, so we know there exists $x,y \in \mathbb{Z}$ such that $d = ax + by$, and ...
0
votes
0answers
31 views

Prove that $s \not= \emptyset $ by showing that at least one of $|a|$ or $|b|$ is an element of $S$.

. I'm trying to show an alternative proof to Bezout's Lemma (let $a, b \in \mathbb{Z}$, then there exists $x,y \in \mathbb{Z}$ such that $gcd(a, b) = ax + by$). Heres one of the steps in proving it: ...
0
votes
2answers
34 views

Proof that $t^ne^{-t}\leq Ce^{-t/2}$ for all $n\geq 1$ and $t\geq 0$

How do I prove that $t^ne^{-t}\leq Ce^{-t/2}$ for all $n\geq 1$ and $t\geq 0$. I am not sure which type of proof to use, for example induction with two variables. The graphs suggest C can always be ...
1
vote
1answer
68 views

Possible book correction or am I missing something?

Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
0
votes
1answer
34 views

Prove something that is differentiable

The question states If g(x) is differentiable, then for any positive integer $n$, $(g(x))^n$ is differentiable and $\frac d{dx}$$(g(x))^n=(g(x))^{n-1}g'(x). $ Where does the continuity of g enter ...
0
votes
1answer
46 views

Basic Field Properties: multiplication

I am struggling with the proofs: a) $(a^{-1})^{-1} = a$ b) $(-a)^{-1} = -a^{-1}$ I have done the rest of the theorem but it is just these two that are difficult. To prove them you can only use the ...
17
votes
5answers
518 views

Proving a certain map on the closed unit disc must be the identity

Bounty expired. Will gladly re-create one if a satisfactory answer is posted in the future. Prove: Let $f$ be a continuous function on the closed unit disc with two properties: 1. $f$ is the ...
1
vote
2answers
46 views

Let $F = \{2x-3:x \in E\}$. Show that $F$ is compact.

Suppose that $E$ is a compact nonempty subset of $\mathbb{R}$. Let $F = \{2x-3:x \in E\}$. Show that $F$ is compact. My idea is to prove that $F$ is closed and bounded. To prove that it is closed, ...
1
vote
3answers
75 views

Show that $(x_n)$ is decreasing and find its limit.

Let $0<x_1<1$. For $n \in \mathbb{N}$, let $x_{n+1}=1- \sqrt{1-x_n}$. Show that $(x_n)$ is decreasing and find its limit. I did: $$x_{n+1} = 1- \sqrt{1-x_n}$$ $$x_{n+1} - x_n= 1- \sqrt{1-x_n} - ...
1
vote
2answers
86 views

How do I prove that $d(x,y)=(|x-y|)^{\frac{1}{2}}$ is a metric?

Let $X$ be a metric space with metric defined by $$d(x,y)=\sqrt{|x-y|}$$ where $x, y\in X$. How do I prove the triangle inequality for the metric $d(x,y)=\sqrt{|x-y|}$?
0
votes
2answers
62 views

How do I show that $f(x)=x^2 + x$ is uniformly continuous on $(0,1)$?

I know how to show that $f(x)=x^2$ is uniformly continuous, but I am confused when it is $x^2 +x$
1
vote
2answers
45 views

prove that for $n \ge 4, {{2n}\choose{n}} \ge n\cdot2^n$

Prove that for $n \ge 4$ $${{2n}\choose{n}} \ge n\times2^n$$ I tried like that: $T_4$: ${{8}\choose{4}} = 70 \ge 4\times2^4$ = 64 so it's ok $T_{n+1}$: $$\frac{(2n+2)!}{(n+1)!)(n+1)!} \ge ...
0
votes
1answer
78 views

Proving the formula for the directional derivatives of the of the sum and dot product of two functions

Define the directional derivative of a function $\textbf{f}$ at $\textbf{c}$ in the direction $\textbf{u}$ by $$\textbf{f}\hspace{0.04in}'(\textbf{c};\textbf{u}) = \lim_{h \rightarrow 0} ...
0
votes
1answer
43 views

Does $ \log(x)^{x^a}$ eventually dominate $x^k$?

Does $ \log(x)^{x^a}$ eventually dominate $x^k$ for all $a\gt 0$ and for all positive integers $k$? And if so, how does one prove this? Thanks a lot for your help.
2
votes
1answer
40 views

Functions, Continuity and IVT

Suppose that $g$ is a function defined and continuous on $\mathbb{R}$ and $n$ is a positive integer such that $$\lim_{x\to \infty} \dfrac{g(x)}{x^n} = 0 = \lim_{x\to -\infty} \dfrac{g(x)}{x^n}$$ (i) ...
0
votes
2answers
44 views

Functions and the IVT

Let $g, h$ be continuous functions defined on some interval $J$ and suppose that $g(x) \neq 0$ for any $x \in J$. If $g(x)^2 = h(x)^2$ for all $x \in J$, show that either $g(x) = h(x)$ for all $x \in ...
1
vote
2answers
36 views

Limit and maximum: IVT

Let $f$ be a function defined and continuous on $\mathbb{R}$. Assume that $f(a) > 0$ for some $a \in \mathbb{R}$ and that $$\lim_{x\to \infty} f(x) = 0 = \lim_{x\to -\infty}f(x)$$ Show that ...
1
vote
2answers
203 views

Limit of functions and the Binomial Theorem

If $n \geq 2$ is an integer, show $n^{1/n} = 1 + h$; where $h \leq \sqrt{ \dfrac{2}{n-1}}$ Then Deduce that: $\lim\limits_{n \to \infty} n^{1/n} = 1$ Hint: Since $n>1$, $n^{1/n}>1$. So, ...
1
vote
2answers
46 views

Limit of Fuctions

Let $f(x)= \left \{ \begin{array}{cc} x & x\in \mathbb{Q}\\ 0 & \,\,\,\,\,\,x\in \mathbb{R}\setminus\Bbb{Q} & \end{array} \right . $ Determine all $a \in \mathbb{R}$ for which ...
0
votes
3answers
68 views

Cauchy convergent sequences

Suppose that $(a_n)$ and $(b_n)$ are convergent sequences and that $b_n > 0$ for all $n$. Is it true that $(a_n / b_n)$ is Cauchy? If it is true, prove it. If it is not true, give a counterexample ...
0
votes
2answers
61 views

Sequence and convergence of subsequences

Suppose that $(a_n)$ is a sequence. Assume that both $(a_{2n})$ and $(a_{2n+1})$ converge to the same $L$. Prove carefully that $(a_n)$ also converges to $L$ I was thinking that $(a_{2n})$ and ...
2
votes
1answer
32 views

Prove that $(|u-s|+|x-y|)^2\leq 2|u-s|^2+2|x-y|^2$.

Prove that $(|u-s|+|x-y|)^2\leq 2|x-y|^2+2|u-s|^2$. My professor used this inequality for a proof last week. How would one prove this? I thought about using the Cauchy-Swartz inequality. This is ...
1
vote
1answer
76 views

Prove Heine-Borel Thm

Prove Heine-Borel Theorem: "A subset $S$ of $\mathbb{R}$ is compact if and only if every open cover for $S$ has a finite subcover." Suggestions: Let $S \subset \mathbb{R}$. If every open cover for ...
2
votes
1answer
149 views

Open Cover for a Compact Subset

I am doing some extra exercises for an Analysis class, and I found this one. We haven't seen much of what an open cover is, but I want to learn it. So, here it goes, and thank you everyone! Let ...
1
vote
1answer
234 views

Open cover with no finite subcover

Let (x_n) be a sequence, let $L$ ∈ R, and for each ϵ>0, {k ∈ N : x_k ∈ B($L$; ϵ)} Suppose S is not a compact subset of R. There is some ϵ_L > 0, such that {k ∈ N : x_k ∈ B($L$; ϵ_L)} is finite. ...
0
votes
1answer
58 views

Convergent subsequence

1) Let (x_n) be a sequence and let L ∈ R. Suppose that for each ϵ > 0, {k ∈ N : x_k ∈ B(L; ϵ)} is infinite. Show that (x_n) has a subsequence converging to L. ...
1
vote
1answer
87 views

Where to find a proof of Silverman-Toeplitz?

I am referring to the theorem which gives a necessary and sufficient condition on a infinite matrix that maps convergent sequences to sequences converging to the same limits. Wiki gives a link to ...
0
votes
2answers
39 views

$A \subseteq (X,d)$ is compact. Which metric $p$ makes $(A \times A,p)$ also compact and $d: (A \times A,p) \rightarrow [0,\infty)$ continuous?

$(X,d)$ is a metric space. And $A \subseteq X$ is a non-empty compact set in the metric space $(X,d)$. Then, does there exists a metrics $p$ and if so which metrics $p$ make $(A \times A,p)$ compact ...
0
votes
2answers
20 views

Formally prove: $\lim_{n\to\infty}x_n=L_1\Longrightarrow\lim_{n\to\infty}x_{n+k}=L_1,\forall k\in\mathbb{N}$

OK, so I'm given the following: $$\lim_{n\to\infty}x_n=L_1\iff\forall\epsilon>0,\exists N(\epsilon)\in\mathbb{N}\ni\forall n>N(\epsilon),\ \left|x_n-L_1\right|<\epsilon$$ I just have no ...
2
votes
1answer
174 views

Sufficient and necessary conditions to get an infinite fiber $g^{-1}(w)$

I want to verify the proof of this result and get some start ideas to overcome the different steps of this proof. Lemma: Let $g$ be a real analytic function. Then we have the equivalence ...
1
vote
2answers
48 views

Does $|\textbf{x}-\textbf y|<\delta$ imply $|x_1- y_1|<\delta$ and $|x_2- y_2|<\delta$

I want to say that $|\textbf{x}-\textbf y|<\delta$ implies $|x_1- y_1|<\delta$ and $|x_2- y_2|<\delta$ for a proof I am working on. This is assuming that $\textbf{x}=(x_1,x_2) \in \text R^2$ ...