0
votes
2answers
22 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
0answers
25 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
41 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
39 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
63 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
37 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
28 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
33 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
62 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 ...
1
vote
0answers
31 views

If $(\sqrt{5}-1)/2 = \sum_{k=1}^{\infty}2^{-n_k}$ where $n_k \in \mathbb{N}$ then $n_k \leq 5\cdot 2^{k-1}-1$

Show that if $(\sqrt{5}-1)/2 = \sum_{k=1}^{\infty}2^{-n_k}$ where $n_k$ are positive integers, then $n_k \leq 5\cdot 2^{k-1}-1$. This is a problem from the book "Problems in mathematicaly analysis" ...
0
votes
1answer
33 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
44 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
491 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
69 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
58 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
61 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
44 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
56 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
39 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
37 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
43 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
34 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
146 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
61 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
57 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
72 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
138 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
202 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
48 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
56 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
37 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
18 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
169 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 $\textbf{x}=(x_1,x_2) \in \text R^2$ ...
0
votes
1answer
74 views

Question about intervals and infima/suprema

Let $L(E)$ be the set of lower bounds of a set $E$ and $(S, \le)$ a partially ordered set. For each $s \in S$, let $$ \langle s] := \{x \in S \mid x \le s\} $$ and $$ [s\rangle := \{x \in S \mid ...
2
votes
1answer
107 views

Problem of proofs

I've been away from math for a long time ,and while I was trying to relearn it using Courant and Fritz 's booknon calculus,I loved the explanations but I couldn't solve any exercices(they're almost ...
0
votes
1answer
156 views

Showing triangle inequality for a norm

I want to determine whether the following is a norm or not: \begin{equation} ...
3
votes
1answer
227 views

Can somebody correct my proof? : if f is continuously differentiable, then f is differentiable.

Theorem: Let $U\subseteq \Bbb R^n$ be open. If $f$ has continuous first partial derivatives in $U$ then $f$ is differentiable in $U$. Proof: Let's prove that if $f$ is differentiable at $a$ ...
0
votes
1answer
40 views

Showing a function is contractive

This seems to simple of a question and thus I am doubting myself... Show that the function $\dfrac{1}{2}x$ on $1\leq x \leq 5$ is contractive. \begin{align} |F(x) - F(y)| =& \left|\dfrac{1}{2}x ...
0
votes
0answers
72 views

Cauchy sequence proof problem.

I worked out the following problem Show that $\{a_n\}$ and $\{b_n\}$ are equivalent Cauchy sequences iff $\{c_n\} = \{a_1,b_1,a_2,b_2, \cdots \}$ is Cauchy. my proof: $(\Leftarrow)$ Suppose ...
0
votes
4answers
313 views

All real numbers can be expressed as a limit of rational numbers?

RTP Let $C$ be a set of Cauchy sequences. $\forall x \in {\Bbb R}, \exists \{a_n\} \in C$ sucht that ${a_n} \to x$. I have no clue to even start this problem. All I know so far is that $\Bbb R$ ...
13
votes
2answers
585 views

Where's the error in this $2=1$ fake proof? [duplicate]

I'm reading Spivak's Calculus: 2 What's wrong with the following "proof"? Let $x=y$. Then $$x^2=xy\tag{1}$$ $$x^2-y^2=xy-y^2\tag{2}$$ $$(x+y)(x-y)=y(x-y)\tag{3}$$ ...
1
vote
2answers
131 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 ...
1
vote
0answers
103 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 ...
3
votes
0answers
33 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
0answers
161 views

Bounds for the exponential integral

In Abramowitz and Stegun: Handbook of Mathematical Functions (on page 229, property 5.1.20) it is found that $$ \frac{1}{2} \log \left(1 + \frac{2}{x} \right) < \exp(x) E_1(x) < \log \left(1 + ...