Tagged Questions

0
votes
1answer
31 views

Correctness of Analysis argument with Cauchy sequences

Let $(x_n)$ and $(y_n)$ be Cauchy sequences in $(X, d)$. Show that $(x_n)$ and $(y_n)$ converge to the same limit iff $d(x_n, y_n) \rightarrow 0$ Proof $\rightarrow$ Suppose $(x_n) \to a$ and $(y_n) ...
4
votes
2answers
254 views

A continuous function $f$ from a closed bounded interval $[a, b]$ into $\mathbb{R}$ is uniformly continuous

I am reading (as a supplement) the book Basic Real Analysis, by Anthony Knapp. Before I proceed into reading a proof, I want to be sure that the result seems obvious. Yet, I am having trouble seeing ...
2
votes
1answer
106 views

Limit of $\arctan z$

,$\displaystyle \lim_{z \rightarrow \infty} \arctan(z) = \frac{\pi}{2} $. One way to see this is to put $\displaystyle z = \frac{y}{x}$ and imagine $y$ and $x$ as the sides of a right triangle. Then ...
1
vote
3answers
131 views

Show that there exists a positive real number $x$ such that $x^3 = 5$.

Here is what I've done so far: [First, want to show $b = 5$ is an upper-bound of $S$.] So, let: $$S = \{x \in \Bbb R : x \gt 0, x^3 \le 5\}, S \neq \emptyset$$ Assume that $b = 5$ is not an ...
0
votes
1answer
73 views

Simple proof involving inequality [duplicate]

Possible Duplicate: Proof with inequalities I've just started reading a book on real analysis and a lot of my proofs reduce to proving this fact over and over again: For all $\epsilon ...
3
votes
2answers
183 views

I can't find the contradiction in this proof

The problem is "You have a function $f:(a,b) \rightarrow \mathbb{R}$. $\exists u \in$ $\mathbb{R}$ such that $f(x) < u \forall x \in (a,b)$. Prove that if the limit of f(x) as x approaches b ...
4
votes
4answers
177 views

Proof of the irrationality of $\sqrt{3}$ - logic question

Prove $\sqrt{3}$ is irrational. (Proof by contradiction). Let $\sqrt{3}$ be a rational number in simplest form $\frac pq$. So squaring both sides of $\sqrt{3}=\frac pq$ we get $3=(\frac {p}{q})^2$ ...
0
votes
2answers
75 views

Proof for Convergent Sequences [duplicate]

Possible Duplicate: Did I underestimate the limit proof? Let $(a)_{n\in \Bbb N}$ and $(b)_{n\in \Bbb N}$ be sequences of real numbers such that $a_n$ $\le$ $b_n$ for all $n\in \Bbb N$. ...
2
votes
3answers
398 views

Is the set of all positive real numbers dense in $\mathbb{R}$

I am working on a problem I found that asks whether the set $S = \{x \in \mathbb{R} \mid x \ge 0\}$ is dense in $\mathbb{R}$. The theorem I have been using states the following: "$S$ is dense in ...
0
votes
0answers
46 views

Prove existence of equilibrium, solution

I have two questions. 1) When does the product of two non-linear concave functions will also be concave? 2) I have the following equation: $$\frac{0.5\gamma\eta f(x-y^*)}{h''(y^*)-\eta ...
2
votes
2answers
271 views

Spivak Calculus 3rd ed. $|a + b| \leq |a| + |b|$

I'm working through the first chapter of Michael Spivak's Calculus 3rd ed. Towards the end of the chapter he proves $ |a + b| ≤ |a| + |b| $ using the observation that $|a|= \sqrt{ a^2 }$ when $a$ ...
5
votes
3answers
167 views

How to show x and y are equal?

I'm working on a proof to show that f: $\mathbb{R} \to \mathbb{R}$ for an $f$ defined as $f(x) = x^3 - 6x^2 + 12x - 7$ is injective. Here is the general outline of the proof as I have it right now: ...
4
votes
1answer
510 views

Proving that a metric space is compact

Let $H^\infty$ be the set of real sequences such that each element in each sequence has $|a_n|\leq 1$. The metric is defined as $$d(\{a_n\}, \{b_n\}) = \sum_{n=1}^\infty \frac{|a_n - b_n|}{2^n}.$$ ...
7
votes
0answers
315 views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
1
vote
1answer
476 views

Proof that a sequence is divergent

Is my proof valid? Suppose $a_n=\frac{1}{n^2}-\sqrt{n}$. I will now proof that $a_n$ diverges: $\forall \epsilon > o \exists n^* \forall n \ge n^*: |a_n - L| < \epsilon$ So ...