0
votes
2answers
64 views

Showing $a_n=\sin(n)$ does not converge

Show that $a_n=\sin(n)$ does not converge My idea: Take two subsequences: $a_{n_k}=\sin(\frac {\pi k} 2)$ , $a_{n_l}=\sin(\frac {2\pi l} 3)$ So: $\forall n$ : $\lim_{n\to\infty} a_{n_k}=1$, ...
0
votes
0answers
11 views

Deducing the rules for inequalities from the order properties.

I'm trying to relearn calculus but going more in-depth this time, meaning doing all exercises and especially the ones dealing with proofs. I'm using the Thomas calculus textbook. Thomas starts by ...
0
votes
2answers
45 views

Using epsilon -delta definition to show that $\lim_{x \rightarrow 2} (x^3 + \sqrt[3]{x}) = 8+ \sqrt[3]{2} $

I'm trying to use the $\varepsilon$-$\delta$ definition to show that $\lim\limits_{x \rightarrow 2} (x^3 + \sqrt[3]{x}) = 8+ \sqrt[3]{2} $. I already know how to prove continuity for cubic root using ...
1
vote
2answers
54 views

l'Hopitals rule - is my working correct?

Is anyone able to help me with this question on l'Hopital's rule? Use l'Hopital's rule to find the limit of the sequence $\{a_n\}_{n=1}^\infty$ with $n$-th term $\displaystyle a_n = ...
0
votes
0answers
34 views

Proof verification problem

I would like to know what is the true output, and what is the way of solving it? To me, I have got solution to be exact as Q1.
0
votes
2answers
31 views

Anyone have a good proof for the second part of FTC?

Does anyone have a good proof for the second part of the fundamental theorem of calculus? I haven't been able to find any good videos on it so far so I'd like someone to write it down and I can throw ...
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" ...
1
vote
2answers
43 views

Geometrical Application of Calculus with Speed

Two vehicles are heading for a crossroad (point $C$) and intend to pass straight through. Vehicle $A$ is $100\,\mathrm{km}$ due North travelling at $80\,\mathrm{km}/\mathrm{hr}$ towards $C$ Vehicle ...
1
vote
1answer
29 views

“Identity” of volume of a solid of revolution (Limted Weierstrass function)

The question is: Is this correct? This is pretty much the first thing I've tried to come up with by myself. I wanted to see what would happen if I tried to calculate volume of a solid of revolution ...
1
vote
0answers
44 views

check my answer - Show that $f(A)=trace(A^2)$ is differentiable and find the differential at any point

As topic says, we are given $f: Mat_n(\mathbb R) \to \mathbb R,f(A)=trace(A^2)$ where $A$ is an n by n matrix with real entries. I think I managed to show that $f$ is both differentiable, and find ...
3
votes
4answers
121 views

Proving the limit $\lim_{x \to 2}\frac{1}{x}=\frac{1}{2}$

I wish to prove the limit $$\lim_{x \to 2}\frac{1}{x}=\frac{1}{2}$$ In other words, given $\epsilon > 0$, I wish to prove that I can find a $\delta > 0$ so that $$|x - 2| < \delta \implies ...
0
votes
2answers
77 views

Using Darboux Sums to Prove Upper and Lower Integrals

Define $f:[0, 1]\rightarrow\mathbb{R}$ as \begin{equation} f (x) \equiv \left\{\begin{array}{l l} x & \text{if } x\in [0, 1]\cap \mathbb{Q}\\ 0& \text{if }x\in [0, ...
0
votes
1answer
103 views

Comparison Theorem for Integrals

Problem: Let $a>0$ and $b>a+1$. Use the Comparison Theorem to show that the following integral is convergent: $$\int ^ \infty _0 \frac{x^a}{1+x^b} \ dx$$ My attempt at this was that since ...
5
votes
3answers
102 views

Anything wrong with this 'proof'?

Problem. Show that $e^{-x}$ and $\sin(x)$ intersect infinitely many times. Solution. $\lim_{x \to o} e^{-x} = e^0 = 1$ $\lim_{x \to \infty} e^{-x} = 0$ This shows that as $e^{-x}$ goes from 0 to ...
3
votes
2answers
82 views

If $f$ is differentiable and $\lim_{x→0} f'(x) = L$, then $f'(0) = L$.

True/False. (c) If $f$ is differentiable on an interval containing zero and if $\lim_{x→0} f'(x) = L$, then $f'(0) = L$. 1. How to presage proof by contradiction? Proof by contradiction. ...
2
votes
1answer
35 views

Does all non-monotonic continues functions have $x_0 \in \mathbb{R}$ such that $f'(x_0)=0$?

Given $f\colon\mathbb{R} \to \mathbb{R}$, $f$ is differentiable on $\mathbb{R}$ and the $\lim_{x \to \infty}f(x)$ does not exists . show/prove formally that there exists $x_0 \in \mathbb{R}$ such ...
7
votes
0answers
107 views

My proof of the First Fundamental Theorem of Calculus

I've tried to prove the theorem in advance at the level that satisfies me. The notation used might not be correct, but I hope all major steps are correct. DEFINITION (from Apostol's Calculus I): Let ...
0
votes
1answer
104 views

help and verification of 3 short exercises

I'm reading an old book and find the following three question. I'd like to know two things: if my attempts are correct and also it would be great if someone could give suggestions in more detail. ...
0
votes
1answer
42 views

Is this proof circular? (Proof that the open disk/ball of radius 'r' is an open set)

This is on page 109 of "Vector Calculus" (5th ed.) by Marsden and Tromba. Here's my problem with this proof: We're trying to prove that an Open Disk is, in fact, an open set, but when we use the ...
1
vote
1answer
45 views

A rearrangement of an absolutely convergent complex series is also absolutely convergent

I just completed the following proof. Is it valid? Let $\sum_{k=1}^{\infty} a_k$ be an arbitrary convergent series that also converges absolutely. Then $\sum_{k=1}^{\infty} a_k \in \mathbb{C}$ and ...
2
votes
0answers
63 views

Evaluation of $A$ in $2K(\sqrt{x}) = -\log(1 - x) + A + o(1)$ when $x \to 1^{-}$

Let $$K(k) = \int_{0}^{\pi/2}\frac{dx}{\sqrt{1 - k^{2}\sin^{2} x}}$$ be the complete elliptic integral of first kind where $0 < k < 1$. Let $k' = \sqrt{1 - k^{2}}$ be the complementary modulus. ...
1
vote
2answers
85 views

prove or give a counter-example

I think I have solved it (please check) but I would like to see and (re)-learn how one writes a proper proof (including the mathematical signs) and little things (I might have missed), maybe even more ...
0
votes
1answer
41 views

Check my proof of the “Boundedness theorem”

Theorem: Let $f$ be continuous on a closed interval $[a, b]$. Then f is bounded on $[a, b]$. Proof (sketch): Suppose $f$ is unbounded. Let's define the set $N$ containing those $x$ for which $f$ is ...
3
votes
0answers
91 views

My proof of Bolzano's theorem

Before I read the proof of Bolzano's theorem from my Calculus book, I've tried to prove it myself. I will use the following lemma and the least upper bound axiom. [Lemma: Sign-preserving property of ...
4
votes
1answer
214 views

Problems with fake proofs of limit of sequences

I can hardly imagine an easier example of the fact that my understanding of the topic is more than rusty. I will divide the question in two parts to make the reading easier: 1) Background; 2) ...
1
vote
0answers
85 views

Is this a correct way to prove what the derivative of a polynomial function is?

After trying a polynomial long division problem with a lot of wondering how to go about answering it I proceeded by most likely overcomplicating things but the equation derived seems to work at ...
0
votes
1answer
20 views

Can I use this proof that $\lim_{x\to p}[1/f(x)] = 1/\lim_{x\to p}f(x)$?

My textbook on Calculus use a much wordy proof. Maybe the author didn't want to declare that $f(x)/f(x) = 1$ since we have not prooven it. And maybe there is more in this statement that meets the eye. ...
3
votes
2answers
96 views

Question about sup norm

Let $x \in \mathbb{R}^n$. Define $|x| = \max\{ |x_1|,...,|x_n|\} $. I want to show that this is a norm on $R^n$. This is my reasoning. First, notice $$ |x| = \max\{ |x_i| \} \geq |x_i| \; \forall i ...
2
votes
1answer
42 views

If $|\nabla F| > 1$ and $|F| \le 1$, is there a zero nearby?

I saw this claim, stated without much explanation, in an article I'm reading: Let $F:\mathbb{R}^n\to\mathbb{R}$ be a $C^1$ function which satisfies $|\nabla F|>1$ everywhere. We know that ...
1
vote
0answers
114 views

Show $\ln(x)$ continuous at $x = 1$

For $\ln x $ to be continuous at $x = 1$ we need to show that for all $\epsilon > 0$ there exists a $\delta > 0$ such that $ 0 < |x - 1| < \delta \implies |\ln x - \ln 1 | < \epsilon $. ...
1
vote
0answers
98 views

$f_1,f_2 :\mathbb{R}^2\rightarrow \mathbb{R}^2$ be two functions such that…

Let $f_1$ and $f_2$ be two functions on $ \mathbb{R}^2$ defined as : $$f_1(x,y)=(x+1,y+3)\\ f_2(x,y)=(x-3,y-2)$$ Which of the following are true? For any positive integer $k$ there exists a ...
0
votes
1answer
47 views

Does this constitute sufficient proof? [duplicate]

Task I have the following function $f(x)=x^2+1$ I need to prove, according to the $\epsilon - \delta$ definition of a limit, that $f(x)$ is continuous at $x = 2$. Step 1 $\forall \epsilon > 0 ...
1
vote
0answers
44 views

Non-negative, continuous function with integral [duplicate]

Let there be an integrable, non-negative function $f$ in a range $[a,b]$. If the integral $\int_a^b f(x) \, dx$ equals $0$, prove that $f(x)=0$ for every $x$ for which $f$ is continuous. I have ...
1
vote
3answers
214 views

Trouble understanding math proofs

*edit Even though there are already answers to my question, I appreciate anyone that offers their advice! I am not sure if this is the right place to ask this but I usually ask for help here. I am a ...
2
votes
3answers
66 views

Proof that the limit of a sequence is equal to the limit of its partial sums divided by n

Let $\{ x_n \}_n$ be a sequence of real numbers. Suppose $ \lim_{n \to \infty}x_n=a.$ Show that $$\lim_{n \to \infty} \frac{x_1+x_2+...+x_n}{n}=a$$ As it is my first proof I'm not really sure ...
1
vote
0answers
62 views

alternate proof of theorem 1.21 of Baby Rudin

I wanted to ask if anyone has tried to prove theorem 1.21 from Baby Rudin book (on existence and uniqueness of positive real n-th root of a positive real number) differently and would care to check my ...
1
vote
1answer
68 views

Review of solution: Prove $\liminf({a_n}) \ge \liminf({b_n})$

${a_n} \ge {b_n}\forall n \in $ Prove: $\liminf({a_n}) \ge \liminf({b_n})$ I proved it by contradiction. Let's assume $\liminf({a_n}) < \liminf({b_n})$. $a := \liminf({a_n})$ $b := \liminf({b_n})$ ...
1
vote
2answers
70 views

Proof that the derivative of a linear function is $0$.

We have a function $f(x)=x$ which is defined and is continuous on the set $S$ of all real numbers. The derivative at point $x$, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}h$$ Using the theorems of ...
2
votes
2answers
68 views

Proving the derivative is $0$ at the extremum and all derivatives are $0$.

The pictures below show the proof that Apostol uses in his book. I can't understand why Apostol introduces the function $Q(x)$ and proves the theorem by contradiction using the sign preserving ...
2
votes
1answer
209 views

Proving that $\sin(x)$ is continuous at $0$

Given: $|\sin x| < |x|$, valid for $0<|x|<\frac12\pi$ (EDIT: $\frac12 \pi$ not $\frac1{2\pi}$) Conclusion: $\lim_{x\to 0}\sin(x) = 0$, also expressed as $|\sin(x)| < \epsilon$ if ...
0
votes
2answers
115 views

Proving that $1/x$ and $1/x^2$ limit does not exist

1) If I am to prove that limit of $ \frac1x$ doesn't exist at $x\to0$ is it sufficient and rigorous enough to show that the left hand and the right hand limits are not equal(EDIT: are not equal ...
1
vote
2answers
73 views

Requesting feedback on proof of theorem

I'm trying to self study my way through Apostol's calculus and have just started. Having completed an undergraduate degree in physics some time ago, the math courses I took mostly focused on ...
1
vote
2answers
216 views

Using inequalities and limits

Is it possible to say: $$ If \ f(x) \ and \ g(x) \ both \ have \ limits \ as \ x\to p\ and \ f(x) \le g(x), \ then \lim_{x \to p} f(x)\le \lim_{x \to p} g(x). $$ My proof(Edit: Proof is wrong due to ...
1
vote
2answers
205 views

Proving the limits of the sum of two functions is equal to the sum of the limits

I am new to proving in math so I want to know if this informal proof of limits is possible: Theorem: If $\lim_{x \to a}f(x)=A$ and $\lim_{x \to a}g(x) = B$, then $$\lim_{x \to a}[f(x)+g(x)]=A+B$$ ...
2
votes
1answer
104 views

Proof verification needed for interesting advanced calculus problem.

let $f:(0,\infty) \rightarrow \infty$ have the following properties: (I suppose $f$ continous) a.) $\lim_{x \rightarrow \infty} \dfrac{f(x)}{x^k}=a ,a \in \mathbb R \bigcup \infty$ b.) $\lim_{x ...
1
vote
2answers
107 views

Proof for a function being integrable

Question: Let $f:[0,1] \to \Bbb R$ be a function s.t. $ \begin{cases} 1 & x=\frac 1n \\ 0 & \text{otherwise}\end{cases}$ prove that $f$ is integrable and that $\int _0^1 ...
5
votes
1answer
124 views

Any continuous group homomorphism $\mathbb{R}\to \mathbb{R}^n$ is $C^\infty$

Show that any continuous homomorphism $\mathbb{R}\to \mathbb{R}^n$, with respect to the usual abelian group structure, is actually $C^\infty$. My attempt: Let $\varphi$ be such a map. $$\lim_{h\to ...
3
votes
1answer
89 views

$\int_A f dm \leq 0 $ for all $A$ lebesgue measurable implies $f \leq 0 $ a.e

$$ \textbf{Problem} $$ $\int_A f dm \leq 0 $ for all $A$ lebesgue measurable set implies $f \leq 0 $ a.e $$ \textbf{Solution (Attempt)} $$ We want to show $X = \{ x : f > 0 \} $ is a null ...
2
votes
2answers
75 views

If $f(x)$ is 2x differentiable in $(a,b)$ & $f'(a)=f'(b)=0$, prove that, $\exists\xi $ in $(a,b)$ S.T. $|f''(\xi )|\leq\frac{4(f(b)-f(a))}{(b-a)^{2}}$

Here is my argument (it doesn't feel 100% correct for some reason): By the mean value theorem, there exists $\xi_{1}$ in $(a,b)$ such that, $$f'(\xi_{1}) = \frac{f(b)-f(a)}{b-a}$$ Since, ...
1
vote
3answers
51 views

Prove that if $n \cdot 2^{-t} <0.01$ then $n \cdot 2^{-t} <\frac{1}{101}$

Is the following theorem true? If $n \cdot 2^{-t} <0.01$ then $n \cdot 2^{-t} <\frac{1}{101}$ for $t,n \in \mathbb{N} $. I've tried basic induction but that has led me nowhere, same with ...