Tagged Questions
0
votes
2answers
139 views
Prove that $\int_{-\infty}^{\infty} \sin x \, dx = 0 $
$$\int_{-\infty}^{\infty} \sin x \, dx$$
When I am doing the proof for this, why do i have to split it into
$\int_{-\infty}^a \sin x \, dx + \int_a^\infty \sin x \, dx $?
where a is a constant
0
votes
2answers
49 views
If $E = \{ x \in \mathbb{R}: \sin(\frac1{x}) = 1\}$ then $l = 0$ is a limit point of E
If $E = \{ x \in \mathbb{R}: \sin\left(\frac{1}{x}\right) = 1\}$, then $l = 0$ is a limit point of $E$.
I have a proof here but I don't quite understand a few points, I hope someone can explain it a ...
4
votes
2answers
80 views
Prove that if a set $E$ is closed iff it's complement $E^{c}$ is open
I was wondering if this proof was right.
$\Leftarrow$ Suppose $E$ is closed. Then choose $x\in E^{c}$, then $x\notin E$, and so $x$ is not a limit point of $E$.
Hence there exists a neighborhood ...
0
votes
1answer
31 views
Subsequence proof
"Let $\{a_n\}$ be a sequence. Let $\{b_{n_k}\}$ and $\{c_{n_k}\}$ be subsequences of $\{a_n\}$. Prove directly that if $b_{n_k} \to b$ and $c_{n_k} \to c$ with $b \neq c$, then $\{a_n\}$ does not ...
1
vote
1answer
38 views
Is my proof showing $Q$ is non-measurable complete?
Is my proof valid or complete? if not, what is missing??
Define $R$ on $[0, 1)$. When $x,y\in
[0, 1)$, $xRy \iff y-x$ is a rational number. Then $R$ is an equivalence
relation. Let $Q$ be the set ...
0
votes
1answer
29 views
Bolzano-Weierstrass theorems question
Prove following theorem.
Theorem : If $x$ is a sequence of real numbers that is both bounded and monotone, then $x$ converges.
I know that $x$ is a sequence of real numbers that is both bounded and ...
5
votes
5answers
237 views
Minimum of $x^2 + 3x - 1$ on $[0,1]$ and $[-2,2]$
Consider the problem of finding the absolute minimum of the function $f : [0,1] \rightarrow \mathbb{R}$ that satisfies $f(x)=x^2 + 3x - 1$ everywhere.
Suppose we suspect, by graphical methods, that ...
0
votes
5answers
78 views
Is this a valid proof to show $\frac{n^5}{2^n}$ diverges?
Claim: $a_n = \frac{n^5}{2^n}$ diverges
Let M be arbitrary
Then
$$
\forall \; n \ge \text{max} \big\{ \big[ \sqrt[3]{M} \, \big] + 1 , 3 \big\} \\
n > \sqrt[3]{M} \\
\implies n^3 > M
$$
And ...
3
votes
2answers
43 views
If $x\lt y $ for arbitrary real x and y there exists a real r $r$ such that $x \lt r \lt y$ and hence infinitely many.
If $x\lt y $ for arbitrary real $x$ and $y$ there exists a real r $r$ such that $x \lt r \lt y$
Prove that there is at least one r satisfying this inequality, and hence infinitly many.
I was ...
2
votes
3answers
83 views
Completeness proof of $\ell^p$
Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct:
Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
0
votes
3answers
47 views
Limit of $a_n = -\cos \left( \frac{\pi}{8n-2} \right)$ (proof)
Warning to anyone who stumbles upon this: It is wrong completely and utterly don't use it for reference, thank you Don and Gerry for helping me see this
So my first question is asking whether or not ...
2
votes
1answer
167 views
lim sup inequality proof - is this the right way to think?
I have tried to read many proofs of this but I'm not sure I get it, so please bare with me.
Show that $\lim_{n \rightarrow \infty} \sup (a_n+b_n) \leq \lim_{n \rightarrow \infty} \sup (a_n)+lim_{n ...
2
votes
1answer
108 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 ...
6
votes
3answers
153 views
Proving {$b_n$}$_{n=1}^\infty$ converges given {$a_n$}$_{n=1}^\infty$ and {$a_n b_n$}$_{n=1}^\infty$
Suppose {$a_n$}$_{n=1}^\infty$ and {$b_n$}$_{n=1}^\infty$ are sequences such that {$a_n$}$_{n=1}^\infty$ coverges to A$\neq$0 and {$a_n b_n$}$_{n=1}^\infty$ converges. Prove that ...
1
vote
2answers
50 views
Proving {$b_n$}$_{(n=1)}^\infty$ converges given {$a_n$}$_{n=1}^{\infty}$ and {$a_n + b_n$}$_{n=1}^{\infty}$ converge
Suppose {$a_n$}$_{n=1}^{\infty}$ {$b_n$}$_{n=1}^{\infty}$ are sequences such that {$a_n$}$_{n=1}^{\infty}$ and {$a_n + b_n$}$_{n=1}^{\infty}$ converge. Prove that {$b_n$}$_{n=1}^{\infty}$ also ...
2
votes
4answers
81 views
Can the definition of continuity be said both of these ways?
So if the definition of continuity is: $\forall$ $\epsilon \gt 0$ $\exists$ $\delta \gt 0:|x-t|\lt \delta \implies |f(x)-f(t)|\lt \epsilon$. However, I get confused when I think of it this way because ...
0
votes
1answer
98 views
A Fable Problem in Calculus Class at UCLA by Doug Jungreis
A couple of years ago, I drove up to the Bay Area, which is 400 miles, and I drove fast, so it took me five hours. At the end of the trip , I showed down, because I didn't want to get a ticket, and ...
0
votes
1answer
57 views
Concluding the convergence of a product of series
Let $a(n)$ be a bounded sequence (not necessarily convergent) and assume $\lim b(n) = 0$. Prove that $\lim a(n)b(n) = 0$. Can we conclude anything about the convergence of $a(n)b(n)$ if $\lim ...
1
vote
1answer
111 views
Proving the absolute value of a sequence converges
Prove that if the sequence $\,\{a_n\}\,$ converges to $A$, then $\,\{|a_n\}|\,$ converges to |A|. Also, is the converse true?
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 ...
2
votes
1answer
43 views
Show that $\exists \epsilon$ such that $f_{\epsilon}:=x+\epsilon g(x)$ is one-to-one where $g$ has bounded derivative
Consider a differentiable function $g : \mathbb{R} \to \mathbb{R}$ with bounded derivative $g'$, i.e. $\exists M>0$ such that $|g'(x)|\leq M$ for all $x\in \mathbb{R}$. Prove that for sufficiently ...
4
votes
5answers
252 views
How does one begin to even write a proof?
I'm in my first proof based class and I'm just having a lot of trouble writing proofs. I mean I know it's not going to come natural and it will take time, but seroiusly, how does someone begin to ...
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 ...
1
vote
2answers
144 views
Why is this last step even necessary in this proof with open sets?
Let $E^o$ denote the set of all interior point of a set $E$. Prove that $E^o$ is always open.
Proof: For $p \in E^o$, there is a neighborhood $N_r (p) \subset E$. Since neighborhoods are open, for ...
2
votes
1answer
215 views
Every finite set contains its supremum: proof improvement.
Every finite subset of $\mathbb R$ contains its supremum (and its infimum)
Proof Let $A=\{a_1,...,a_n\}$ be a finite subset of $\mathbb{R}$. Since it is non-empty and it is bounded ($\max A$ is ...
1
vote
2answers
210 views
Evaluation of Derivative Using $\epsilon−\delta$ Definition
Consider the function $f \colon\mathbb R \to\mathbb R$ defined by
$f(x)=
\begin{cases}
x^2\sin(1/x); & \text{if }x\ne 0, \\
0 & \text{if }x=0.
\end{cases}$
Use $\varepsilon$-$\delta$ ...
1
vote
1answer
80 views
Proof simplification and absolute value
Could someone verify the following absolute value inequalitiy:
$b_k < |\epsilon + L| \iff b_k < \epsilon + L $ and $bk < -(\epsilon+L) \iff -bk > (\epsilon +L)$
All together:
$-b_k > ...
1
vote
1answer
66 views
simple statement proof
i am proving this statement about strict isotoneness. i will try on my own and you will tell me whether i am okay or not :)
$A$ is subset of $\mathbb{R}$
$f$ is strict isotone $ \Longleftrightarrow ...
2
votes
2answers
102 views
very simple statement but how to find inf, sup
i am having trouble with this this. i need to simplify this statement and find, if any, the minimum, maximum, infimum, and supremum of it.
the statement is this:
$A:= (]1,2[ \cup ]2,3]) \cup \{2\} ...
3
votes
2answers
378 views
integral property proof
I am having trouble with the following proof:
Prove that if $f$ is differentiable on a closed interval $[a, b]$ then for every continuous function $g$ with the property $\int\limits_a^bf(x)g(x)dx = ...
1
vote
0answers
128 views
The Lebesgue criterion for Riemann integrability with no oscillations
For the Lebesgue Criterion for Riemann integrability, I am trying to prove it without the use of oscillation.
The Lebesgue Criterion for Riemann integrability states that if f:[a,b] is bounded, then ...
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 ...
3
votes
2answers
150 views
Is this proof that $\Delta$ is countable correct?
I have to prove the following
THEOREM Let $f:[a,b]\to\Bbb R$ be such that $\lim\limits_{y\to x}f(y)$ exists for every $x\in[a,b]$. Then the set $\Delta\subset[a,b]$ where $f$ is discontinuous ...
2
votes
2answers
240 views
$f: \mathbf{R} \rightarrow \mathbf{R}$ monotone increasing $\Rightarrow$ $f$ is measurable
Problem. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a monotone increasing function. Show that $f$ is measurable.
Solution.
We know that the set of discontinuites of any monotone increasing ...
2
votes
0answers
118 views
Transcendence of $e$ (proof)
I'm trying to get through the proof of transcendence of $e$ (the base of the natural logarithm) already for a couple of days, but now I got seriously stuck.
Proof is in most sources roughly the same. ...
4
votes
1answer
166 views
Extension of Fatou's lemma
let $X$ be a finite measure space and $\{f_n\}$ be a sequence of integrable functions, $f_n \rightarrow f\text{ a.e.}$ on $ X$.
I want to show if (1) holds, then (2) holds too.
$$\lim_{n \rightarrow ...
0
votes
4answers
94 views
Proof writing: $t_n=s_{n+k}.$ prove $s_n \rightarrow s \iff t_n \rightarrow s$
First prove $s_n$ converges iff $t_n$ converges
$t_n=s_{n+k}.$ prove $s_n \rightarrow s \iff t_n \rightarrow s$
This is obvious but I am not so sure how to write the proof.
Have:
$\forall ...
2
votes
1answer
213 views
What is the meaning of “the lower integral of f from a to b equals the supremum of the lower Darboux sums of P in [a,b]”
I am re-reading lecture notes and I am having a few issues.
Okay, I understand the concept of a partition. You basically have this bounded function $f:[a,b] \rightarrow \mathbb{R}$ and we are ...
0
votes
1answer
126 views
Every non-empty subset of $\mathbb{R}$ bounded above has a largest element
I restarted my analysis book from page 1 trying to relearn everything because I feel like my knowledge is too fragmented. This true false question asks exactly what the title says. I don't know 100% ...
0
votes
2answers
146 views
Closure proof and closure properties.
Is it true that the closure of any set is closed? I am just assuming this fact from the word closure. My whole proof based on this fact
Proof
Let $A_1 =(a_n)_{n \in\mathbb{N}} = \{ a_n :n ...
1
vote
5answers
178 views
How to prove limit of sequence $a_n = \frac{2^n}{n! }= 0$ [duplicate]
Possible Duplicate:
Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.
I'm not sure how to go about solving this. Right now I'm trying to use the squeeze theorem. ...
0
votes
1answer
69 views
Prove $\lim_{x\to p}(f+g)(x)=\lim_{x\to p}f(x)+\lim_{x\to p}g(x)$
The proposition starts with suppose that the limit of $f(x)=a$ as $x\to{p}$ and the limit of $g(x)=b$ as $x\to{p}$.
I know we have to start of with the definition of what the prop starts of with. But ...
4
votes
5answers
304 views
Prove $\lim _{x \to 0} \sin(\frac{1}{x}) \ne 0$
Prove $$\lim _{x \to 0} \sin\left(\frac{1}{x}\right) \ne 0.$$
I am unsure of how to prove this problem. I will ask questions if I have doubt on the proof. Thank you!
1
vote
2answers
2k views
Prove: If a sequence converges, then every subsequence converges to the same limit.
I need some help understanding this proof:
Prove: If a sequence converges, then every subsequence converges to the same limit.
Proof:
Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k ...
0
votes
1answer
105 views
Verifying this proof: $a_n$ is cauchy $\implies a^2_n$ is cauchy
Prove: $a_n$ is cauchy $\implies a^2_n$ is cauchy
Proof:
Let $a_n$ be a cauchy sequence. Then we know:
$\forall_{\epsilon > 0} \exists N_1 s.t. \forall_{n, m} \implies |a_n - a_m| < \epsilon$ ...
0
votes
2answers
50 views
If $c_n=c$ for all $n>0$, then $\lim_{n\to \infty} c_n=c$.
PF: If given $ \epsilon >0 $, let $ N=1$ so whenever $ n>N $ we have
$ |c_n-c|=|c-c|=0 < \epsilon $.
Therefore the limit of $ c_n=c $ as
$ n \rightarrow \infty $ as required.
This is the ...
1
vote
1answer
1k views
If x and y are arbitrary real numbers with x<y, prove that there are infinite real and rational numbers greater than x and smaller than y
I think I got the right answer for the case regarding the real numbers. The problem is to prove for the rational numbers.
Here is my proof for real numbers:
Given an arbitrary real number $h>0$, ...
5
votes
1answer
506 views
Lipschitz Continuous $\Rightarrow$ Uniformly Continuous
The Question: Prove that if a function $f$ defined on $S \subseteq \mathbb R$ is Lipschitz continuous then $f$ is uniformly continuous on $S$.
Definition. A function $f$ defined on a set $S ...
4
votes
1answer
510 views
Proving that a metric space is compact
Let $H^\infty$ be the set of real sequences such that each element in each sequence has $|a_n|\leq 1$. The metric is defined as
$$d(\{a_n\}, \{b_n\}) = \sum_{n=1}^\infty \frac{|a_n - b_n|}{2^n}.$$
...
2
votes
1answer
62 views
Quantified definition of the derivative
How do you quantify:
A function $f:\mathrm{dom}(f) \longrightarrow \mathrm{codom}(f)$ is differentiable at every $x$ contained in $\mathrm{dom}(f)$ if the limit
$$\lim_{h \to ...
