Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, integration through the fundamental theorem of calculus.

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Proving a limit Epsilon Delta definition

How do I find and prove the limit of the following function using the epsilon delta definition. $$\lim_{x \to 0} \frac{2+\sin x}{3-\cos x}x$$
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17 views

Does this recursion/sequence of iterated infinite sums converge?

Let $n,x\in\mathbb{N}$, $\alpha,\beta,\lambda\in\mathbb{R}^+$, where $\alpha,\beta<1$. Does the following sequence converge (and to what)? $s_0=\alpha n+\lambda$ ...
-3
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0answers
38 views

$\int_{a}^{b}{x^nf(x)dx}=0$ for all $n$ [on hold]

Let $f:[a,b]\to \mathbb R$ be a continuous function. Prove that if $\int_{a}^{b}{x^nf(x)dx}=0$ $\forall n\in \mathbb N$ then $f(x)=0$ for all $x\in [a,b]$
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10 views

Difference of two finite Radon measures

Is the difference of two finite unsigned Radon measures still Radon?
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3answers
27 views

Prove: If $\{a_n / n\}$ converges to L which is not 0, then $\{a_n\}$ is unbounded.

Prove: If $\{a_n / n\}$ $\rightarrow$ L $\neq$ 0, then $\{a_n\}$ is unbounded. I've been working on this problem a while, and I've been having trouble relating ${a_n / n}$ and $\{a_n\}$. I know ...
1
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1answer
19 views

Time Series Analysis.Calculate the variance mean and autocorrelation of the time series below.

For the following time series, calculate the mean, varia nce and autocorrelation function: (a) Y_t=5+Z_t+ 0.6Z_t-1
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3answers
71 views

a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$ [duplicate]

Hi need some help with this problem: Assume $f : \mathbb{R} → \mathbb{R}$ is a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$, and $f(0) = 0$. Then $f(α) = α ...
0
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1answer
24 views

convergence of Taylor series on R

I am doing this problem: Let $f\in C^{\infty}(R,R)$ be an infinitely differentiable function on R. Assume that there exist constants $C>0$ and $\rho>0$ so that for any integer $n\ge 0$, ...
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0answers
23 views

$\displaystyle \lim_{n\to \infty}{(\int_{a}^{b}{(f(t))^ndt})^{\frac{1}{n}}}=M$ where $M$ is the sup [duplicate]

Let $f:[a,b]\to \mathbb R$ be a continuous function. Let $\displaystyle M=\sup_{x\in [a,b]}{f(x)}$. Prove that: $\displaystyle \lim_{n\to \infty}{(\int_{a}^{b}{(f(t))^ndt})^{\frac{1}{n}}}=M$ I ...
1
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2answers
32 views

Equivalence of norms proof

This question is from a set of optional, much harder problems from my first year analysis course, but the subject material is norms on $\mathbb R^K$. (c) Show that there exists a constant $C > 0$ ...
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1answer
22 views

What kind of exponential identity is required here and how will it be used? [on hold]

Prove that there exists a real constant $a$ such that $f(x)=\exp(ax)$.
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18 views

Set of local parametrizations of a torus

$\newcommand{\R}{\mathbb{R}}$I would like some help with the following problem: Fix $R > r > 0$ and define $F:\R^3\to \R$ by $$ F(x, y, z) = (R^2 - r^2 + x^2 + y^2 + z^2)^2 - 4R^2(x^2 + y^2). ...
0
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1answer
33 views

Suppose that the sequence of prices{$p_k$} converges to a limiting price $\bar p$. What must $\bar p$ be?

We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. The demand depends on the current price, $D_k = a + b p_k$ and the supply depends ...
0
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2answers
51 views

Question about showing the harmonic series is divergent.

Is it possible to show that the harmonic series is divergent by showing that the sequence of partial sums is a monotone increasing sequence that is unbounded?
2
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1answer
72 views

Convergence of “alternating” harmonic series where sign is +, --, +++, ----, etc.

Exercise 11 from section 9.3 of Introduction to Real Analysis (Bartle): Can Dirichlet’s Test be applied to establish the convergence of $$ 1 - \dfrac12 - \dfrac13 + \dfrac14 + \dfrac15 + ...
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1answer
38 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 ≤ ...
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2answers
32 views

Find the Fourier series of $\sin^3(x)$ on $[-\pi,\pi]$

I'm having trouble integrating $B_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}\sin^{3}t \,\sin(nt)\,dt$.
3
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3answers
116 views

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges. I used the ratio test and eventually came to that: $\frac{n+1}{en}$ which is approximatley equal to $\frac{1}{e}$ which is less than 1. ...
1
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1answer
30 views

prove $\lim_{n\to\infty}\sup\{|f(x) - f_n(x) | : x \in S \} =0$

I understand how the theorem works but how would you prove that a sequence $f_n$ of functions on set $S \subset \mathbb{R}$ converges uniformly iff $$\lim_{n\to\infty} \sup\{|f(x) - f_n(x) | : x \in ...
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2answers
39 views

Accumulation points of a sequence

I was wondering , if a sequence converges does that mean that the sequence have exactly one accumulation point that occurs at it's limit , and if a sequence doesn't converge then it doesn't have any ...
1
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1answer
36 views

If $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$

If a mapping $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$ The mapping $f$ is injective as $f(x) = ...
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1answer
26 views

Meagre and dense sets

We consider subsets of $\mathbb R$. We know the following: If $A$ is meagre then $\mathbb R \setminus A$ is not meagre. (Converse not true.) If $A$ is meagre then $\mathbb R \setminus A$ is dense. ...
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1answer
57 views

What is $\mathbb{R}^\mathbb{R}$

I do not know what is it. $\mathbb{R}$ is the set of real numbers. How come $\mathbb{R}\times\mathbb{R}\times \dotsc $? Thanks.
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1answer
19 views

If $\mu(E_n) < \infty$ for all $n \in \mathbb{N}$ and $1_{E_n} \to f$ in $L^1$, then $f$ is a char. function of measurable set.

Problem: If $\mu(E_n) < \infty$ for all $n \in \mathbb{N}$ and $1_{E_n} \to f$ in $L^1$, then $f$ is a char. function of a measurable set. Attempt: Since $1_{E_n} \to f$ in $L^1$ , there is a ...
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0answers
19 views

A question on differentiability and boundedness

Let $f:R\to R$ be a differentiable function such that limx->inf f'(x)=1. Show that f is unbounded. Here is my try For $\epsilon>0$, there exists an $M$ in reals such that $|f'(x)-1|<\epsilon$ ...
0
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1answer
29 views

German and French word for Basic sequence

Wikipedia says that a sequence $(x_n)$ is a basic sequence iff it is a Schauder basis of its closed linear span. I was wondering whether there is a French or German word for 'basic sequence'? How ...
5
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2answers
121 views

Prove that no function exists such that…

The exercise goes like this: Find a continous function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall c \in \mathbb{R}$ the equation $f(x)=c$ has exactly 3 solutions; Prove that no ...
2
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1answer
37 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
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1answer
44 views

compact open set?

unfortunately I am completely stuck on the follwing question: Given a non compact metric space M that contains a non empty open compact subset, then M is not connected. What examples are there? How ...
0
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1answer
15 views

For a 2 variable function, are there conditions that guarantee you can verify a limit by using only straight line trajectories?

So a recent post gave a nasty 2-variable function: $$f(x,y) = x^2y/(x^4+y^2)$$ and after changing to polar coordinates, you get that the limit is always equal to zero if you hold $\theta$ fixed and ...
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0answers
17 views

Decomposition of a measure into series

Let $M(X)$ be the vector space of all complex regular Borel measures on a compact Hausdorff space $X$,$||\mu ||=|\mu|(X)$. Suppose $\mu , \lambda_n \in M(X),n\in N^+ $,$||\lambda_n||=1$.Since ...
0
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1answer
37 views

Help, check the uniform continuity

(1) $f(x)=sin(1/x)$ on $(0,1]$ ? ( I know it is not uniform continuous on $(0,1)$) (2) $f(x)= xsin(1/x)$ on $(0,1]$? (3) $f(x)=sin(x^2)$ on $[0, \infty)$?
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0answers
35 views

Bound on $f_n'$ implies uniform convergence of $f_n$?

Let $f_n$ be a sequence of functions that converge pointwise to a function $f$. Suppose I know that $|f_n'(x)| \leq C(x)$ where the constant doesn't depend on $n$. How do I conclude that $f_n \to f$ ...
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0answers
22 views

Some Continuity Question

Suppose $f(x)$ and $g(x)$ are continuous functions on $[a,b]$ with $f$ monotone increasing. Assume there exists a sequence $x_n \in [a, b]$ such that for all $n \in \mathbb{N}$ , $g(x_n) = ...
1
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1answer
23 views

Something basic; why do I get two different bounds on $f(x) = \frac{x^2}{\sqrt{x^2 + n^{-1}}} + \sqrt{x^2 + n^{-1}}$?

Let $n$ be a natural number. Let $f(x) = \frac{x^2}{\sqrt{x^2 + n^{-1}}} + \sqrt{x^2 + n^{-1}}$. since $x^2 + n^{-1} \geq x^2$, it follows that $$|f(x)| \leq \frac{x^2}{|x|} + \sqrt{x^2 + ...
0
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1answer
10 views

Continuous piecewise smooth function $=$ a globally $\mathcal{C}^1$ function $+\sum a_i|s-\alpha_i|$?

I'm reading Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, written by Franck Boyer and Pierre Fabrie. They stated that Such a piecewise smooth ...
0
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3answers
32 views

Convergance of a sequence [on hold]

Prove that the sequence $(a_n)$ converges, where$$a_n=\frac {3+n+4{n^2}}{1-n+3{n^2}}$$ for all $n\ge1$
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2answers
104 views

easy question about contraction maps

Let $X$ be a metric space. Suppse $T:X \to X $ is a contraction. I have shown that $T^n$ where $n $ is positive is a contraction: Question: If $T^n$ is contraction for $n > 1$, do we have that ...
4
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1answer
83 views

Exam Question on Real Analysis

For a continuous function $f$ on $[0,1]$, $\int_0^1f^3(x)x^ndx = 0 $ for each integer $n \geq 1$. Prove $\int_0^1f^4(x)dx = 0 $. (where $f^n(x)$ is the nth power of $f$). And hence prove $f = 0$. I ...
2
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1answer
89 views

$ \lim\limits_{x \to +\infty}x\, e^{-x^2}\int_{0}^{x}e^{t^2}dt $

Hello every one Please I need your help for the 3rd question, I tried but i fail every time. for every real $ x $, we put $ f(x)=e^{-x^2}\int_{0}^{x}e^{t^2}dt $. Show that $ f $ is odd of class $ ...
4
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1answer
36 views

Proof of $\displaystyle \lim_{x \to p+}f(x) = l \land \lim_{x \to p-}f(x) = l \implies \lim_{x \to p}f(x) = l$

Let $f:(a,b) \to \mathbb{R}$ and $p \in (a,b)$. In proving the following implication, I am unsure about one step $\displaystyle \lim_{x \to p+}f(x) = l \land \lim_{x \to p-}f(x) = l \implies \lim_{x ...
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0answers
20 views

Question about integration and sets of measure zero

Let $Q \subseteq \mathbb{R}^n$ be a box, $f: Q \to \mathbb{R}$ be bounded, integrable on $Q$. Suppose $g: Q \to \mathbb{R}$ is another bounded function such that $f(x) = g(x)$ for any $x \in Q ...
2
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2answers
104 views

Questions based on Mean Value Theorems

For every $x \geq 0$ prove that $\sqrt{x+1} - \sqrt{x} = \frac{1}{2\sqrt{x+\theta(x)}}$ for $\theta(x) \in (0,1)$. Also prove that $\theta(x) \in (\frac{1}{4}, \frac{1}{2})$ with $lim_{x \rightarrow ...
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2answers
26 views

Equality of Double Limits of Sequence of Real Numbers

Let $a_{mn}$ be a set of real numbers indexed by $m, n \in \mathbb{N}$. Is there an example of such a set $a_{mn}$ where $$ \lim_{n \to \infty} \lim_{m \to \infty} a_{nm} \neq \lim_{m \to \infty} ...
0
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1answer
30 views

Prove the following using the Real Theorem [on hold]

I came across this question in a book and had difficulties in solving it: Let x,y belong to R and a>0 then show that ...
3
votes
2answers
124 views

Real analysis question involving a linear ODE

Where do I start with this one? This question is really quite difficult..
5
votes
1answer
187 views

Exponentials of rational numbers

Does there exist an $$0<x<1$$ such that $$\forall q \in \mathbb{Q^+}$$ $$q^x \in \mathbb{Q^+}$$
0
votes
1answer
22 views

Continuity Function Problem

Suppose f(x) is a continuous function from [0,1] into [0,1]. Show that there exists a point $\xi \in [0,1]$ such that $f(\xi) = \xi$.
0
votes
1answer
28 views

Show that if $a,k\in \mathbb{Z}$ with $0\leq k \leq a$, then $\binom ak=\frac{a!}{k!(a-k)!}=\binom {a}{a-k}$.

I'm reading Ghorpade's A Course in Calculus and Analysis. Given $a\in \mathbb{R}$ and $k\in \mathbb{Z}$, the binomial coefficient associated with $a$ and $k$ is defined by: $$\binom ak = ...
2
votes
0answers
21 views

Laplace transformation questions

Let $f \in L^1(\mathbb{R}).$ Define the Laplace transformation $F:(0, \infty) \to \mathbb{R}$ by $$F(s)=\int_0^\infty f(t)e^{-st}dt.$$ Then (part 1) $\lim_{s \to \infty}F(s)=0$ and (part 2), if $f ...