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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77 views

A function $f(x)$ that Riemann integrable on $[a,b]$.

Define a function $f(x)$ that Riemann integrable on $[a,b]$. Let $$g(x)=\begin{cases} f(x)&\text{if}&x\in[a,b], \\ f(a)&\text{if}&x<a, \\ f(b)&\text{if}&x>b. ...
5
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3answers
107 views

Is there any method to get a finite sum for $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$?

As we can see on Wikipedia, there are some algebraic methods that give us finite sums for the Grandi's series $$1-1+1-1+1-1+1-1+\cdots$$ Let $S$ be the sum of the Grandi's series. Then ...
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1answer
30 views

How can I calculate EMI mentally in my head?

How can I calculate the EMI mentally? For example - 2 million is the loan amount. The ROI per annum is 12% and the tenure is 20 years. What I can make out is the following - ROI per month is ...
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1answer
39 views

How I can solve this equation with respect to $t$

How I can solve this equation with respect to $t$: $$b(2^{((ln(t))/(ln(2)))+2}-1)a^{t}-a^{2t}-1=0$$ where $a$ and $b$ are real numbers.
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40 views

Time to buy a house without a mortgage equation!!

I am looking into a "real world" calculation to calculate the time taken for someone to buy their own home while they rent it. They do this by buying small pieces of the property every month, and ...
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2answers
58 views

Show that the following statement is a theorem.

Suppose h is not a counting number and h is greater than 1, then there is a counting number n such that h is between n and n + 1. I am working through "Creative Mathematics" by H.S. Wall. The book ...
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27 views
+50

Dimension score for metric spaces $\mathbb{R}^n$

Given any metric space $(X,d)$, define its score $S(X)$ to be the smallest value of $k$ such that for every $x\in X$ and $r>0$, the ball $B(X,r)$ is covered by at most $2^k$ balls of radius $r/2$. ...
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1answer
16 views

Sequence inequality does not guarantee limit inequality

Suppose that the sequences {an} and {bn} converge to A and B, respectively, and suppose that there exists n1 such that for all n ≥ n1, we have an < bn. Verify that it is incorrect to conclude that ...
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34 views

A direct proof on if $(X, ||\cdot ||)$ is a normed vector space and $Y\subset X$, with $Y$ having finite dimension, then $Y$ is closed.

I am trying to produce a direct proof on the statement mentioned above. The field I am working in is $\mathbb{R}$. My proof outline goes as following: If $Y$ is finite-dimensional, there exists a ...
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1answer
27 views

On convergence a.e and convergence measure

I have a question. First, I know that convergence in measure of a sequence of functions $f_n$ is different than convergence a.e., wich means there are sequences that converge in measure but not a.e. ...
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1answer
17 views

If f is a real function, continuous at a and f(a) < M, then there is an open interval I contianing a such that f(x) < M for all x in I.

Can someone please help? If f is a real function which is continuous at a ∈ R and if f(a) < M for some M ∈ R, prove that there is an open interval I containing a such that f(x) < M for all x ∈ ...
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45 views

Proof the limit of a closed set exists [on hold]

Let L ⊆ R. The set L is called open if for any x ∈ L there exists e > 0 such that (x−, x+) ⊆ L. The set L is called closed if its complement L c = {x : x /∈ L} is open. (a) Prove that L is closed if ...
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0answers
15 views

Prove that the evaluation map $E_{x_0}: C(K) \to \mathbb{R}$ is differentiable

Let $K \subset \mathbb{R}^m$ be compact, and pick any $x_0 \in K$. Show that the function $E_{x_0}: C(K) \to \mathbb{R}, E(f) = f(x_0)$ is differentiable. For functions between Euclidean spaces, I'm ...
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4answers
33 views

Limit of implicit function

For $v>0$, let $f(v)$ be the smallest positive solution $x$ of $$\sqrt{\left(\frac{v}{x}\right)^2-1}=\tan x.$$ It can be confirmed graphically that $f(v)$ exists for all $v>0$. How can I show ...
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34 views

Prove $f(x)=x$ is Lebesgue integrable on $[0,1]$

Prove that $f(x)=x$ is Lebesgue integrable on $[0,1]$. My definition of integrable comes from Royden's Real Analysis (4th ed). So $f$ is integrable if the lower integral is equal to the upper ...
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1answer
31 views

Temperature defined on a tetrahedron

I am asked to prove that the temperature of a tetrahedron must have at least three distinct points on the edges or vertices of the tetrahedron with the same value. I may assume that the temperature is ...
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2answers
35 views

Real numbers: powers inequality

Trying to prove the following inequality from textbook. Let $x>1$ be a real number, and let $q>0$ be a rational number. Then $x^q>1$. Let $p<q$ be a rationals numbers, and let $x>1$ ...
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1answer
58 views

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? [duplicate]

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? My feeling is that this is not necessarily true. But cannot come up with an example. Can someone provide ...
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1answer
27 views

Verify that $p \circ p=p$ but that $p$ is not self-adjoint.

Consider the following subspaces of $\mathbb{R}^2$: (a) $\{(x,0)\}$ (b) $M=\{(x,y):x=2y\}$. Find an expression and a matrix representation for the oblique projection $p$ of a vector $v \in ...
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1answer
19 views

how to test for that alternating series

$\sum_{n=1}^\infty\frac{(-1)^n(n)}{n+1}$ i did the ratio test for this series and it shows that it is inconclusive because it is equal to 1 and the same thing for root test i guess How to test for ...
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2answers
31 views

Demonstrating the convergence of $x_{n+1} = 1/2(x_n + c/x_n)$

I'm curious how to demonstrate the convergence of such a sequence. It is given as: Let $c > 0$. Define $g(x) = \dfrac{1}{2}\left(x+\dfrac{c}{x}\right)$ when $x \ne 0$. Let $x_0 > 0$, $(x_0)^2 ...
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1answer
42 views

Subsets of real numbers [on hold]

Suppose that O and F are subsets of the set of real numbers. Assume that O is open and F is finite. Does it follow that O\F is open?
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1answer
10 views

Demonstrating the connectedness of the set $A_j = \{(1,0),(0,0)\} \cup \{(x,y) : 0 < y < 1/j\}$

I'm trying to demonstrate the connectedness of the set $A_j = \{(1,0),(0,0)\} \cup \{(x,y) : 0 < y < 1/j\}$. This is for my class in real analysis, so I can't apply concepts that are too ...
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3answers
38 views

If d is a norm on V, is $\frac{d(x,y)}{1+d(x,y)}$ a norm on V?

Let d be a norm on a vector space V and let $\psi:V \to [0,\infty)$ be a function defined as $\psi(v)=\frac{d(v)}{1+d(v)}$. Is $\psi$ a norm on $V$? It seems that $\psi$ does not satisfy the ...
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1answer
12 views

Simplification of a large sum obtained from the 1-D wave equation

I have acquired the sum below through Fourier, and was wondering if there was anyway to simplify it, since it is large and ugly. $$\sum \limits_{n=1}^\infty \frac{-2K_1}{n\pi} ...
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3answers
78 views

If a set $S\subset\mathbb R$ is not closed, does it contain a convergent sequence with a limit outside of $S$?

Suppose S is a subset of R and that S is not closed. Must it follow that there is a convergent sequence in S that converges to some l not in S?
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0answers
28 views

How to show function is measurable if and only if each component is

Let $(X,\mathcal{M})$ be a measurable space and $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ be a measurable space of Borel sets on the real line. Let $f_i:X\rightarrow \mathbb{R}$ be given for ...
4
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1answer
25 views

Calculation of the limit for a composite function

Let's assume that we have a composite function $g(f(x))$, which is not defined at $f(x)=0$. We know that $\lim_{x \to 0} f(x) =0$. Further, we know that $\lim_{f(x) \to 0}g(f(x))=L $. $f(x)$ is a well ...
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1answer
18 views

Proving that any subsequence of a convergent sequence in a metric space is itself convergent

I am doing practice exercises in preparation for a midterm and I am stuck on the following questions (there are in fact two questions, but I thought it would be better to split them apart). I will ...
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2answers
37 views

Convergence of $\{a_n\}$ given that $b_n = a_n + 1/a_n$ converges

I am given a sequence $\{a_n\}$ with $a_n > 0$ and $b_n = a_n + 1/a_n$. I am first asked to assume that $a_n \ge 1$ and show that the convergence of $\{b_n\}$ implies the convergence of $\{a_n\}$. ...
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52 views

Distance function is in fact a metric

I know I should be able to show this, but for some reason I am having trouble. I need to show that $$d(x,y) = \frac{|x-y|}{1+|x-y|}$$ is a metric on $\Bbb R$ where $|*|$ is the absolute value metric. ...
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1answer
23 views

Infinite series and the ratio test

for the series $$\sum_{n=1}^\infty\frac{\sin^2 (n)}{n^2}$$ which convergence test should i apply , i am thinking the comparison test but then to which series i should compare it to . The fact that ...
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2answers
42 views

Determine whether or not $\sum_{k=1}^{\infty} \frac{1}{k- \mathrm{e}^{-k}}$ converges.

I have the following so far Let $a_k = \frac{1}{k- e^{-k}}$. Now, $\lim_{k \to \infty}a_k=0 \implies$ $\sum a_k$ can either converge or diverege. We must thus do further tests to determine whether ...
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32 views

The second derivative of $f^{-1}$ and another question. :)

Suppose both $f$ and $f^{-1}$ are twice differentiable functions. Derive a formula for $(f^{-1})''$. My attempt: We have that by the inverse function theorem that: ...
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1answer
55 views

Why doesn't cantor's theorem work with nested intervals work with rationals?

I understand how and why does the cantor's theorem with nested intervals proof work. I'm also aware that the nested interval property doesn't generally work for $\Bbb Q$ - rational numbers - but I ...
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2answers
24 views

Showing that a multivariable function is one to one

I am stuck with the following problem I am given the function $f$ such that $f(x,y)=(x^2-y^2,2xy)$ I am supposed to show that the function is one to one. For a function to be one to one, $f'>0$. ...
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37 views

Is this integral in its most simplified form?

The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$ it can ...
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Prove the continuity and differentiability of a function in a point. [duplicate]

This question Is the same the question as this one (that I have posted yesterday at 12 am that I why I disconected from a large period of time) Prove that a function is both differentiable and ...
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22 views

Is the sum of a bounded monotone sequence monotone?

So if I have a sequence ${a_n}$ that is bounded and monotone, is $\sum_{j=1}^{n} {a_n}$ monotone?
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22 views

On Compact and Measurable Sets with Positive Length

Greetings fellow Mathematics enthusiasts! The following two-part problem is giving me trouble, and I was hoping someone could help me solve it. It is coming from Terrence Tao's Introduction to Measure ...
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32 views

Closed unit ball in $C[0,1]$ is not compact

How do I prove that the unit ball $B=\{\, f\in C[0,1] :\|f\|_\infty = \max_{x\in C[0,1]}|f(x)|\leq 1\,\}$ is not compact. By taking sequence $ f_n(x) = x^n $ we can prove that $C[0,1]$ is not ...
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0answers
27 views

comparing cardinality of infinte sets [duplicate]

Let's say I have two infinite sets: 1) the set of all functions from $\mathbb{R}$ to {0,1}; 2) the set of all polynomials whose coefficients are in $\mathbb{R}$. Which is greater? I figure the ...
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31 views

example of two continuous real-valued functions whose product is 0

Is there an example of two continuous real-valued functions, say on some interval, whose product is 0?
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1answer
42 views

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge to $w_0$ and $n \rightarrow \infty$? I would say ...
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26 views

Proofs Regarding Open and Closed Sets

I need to prove the following regarding open and closed sets: 1. A set L is closed iff for any converging sequence $(x_n)$ with $x_n\in L$, the limit $x=\lim_{n\to\infty}{x_n}$ is also an element of L ...
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2answers
16 views

In $(]0,1], d_{\mid.\mid})$ why $(\frac{1}{1+n})_{n\in \mathbb{N}}$ do not converge?

Can someone tell me why this sequence do not converge ? First, I know that is a Cauchy's sequence. Then, the fact is that the sequence converges to $0$ when $n \rightarrow \infty$. Thanks in ...
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12 views

Can Banach fixed-point theorem be generalized to the case where each term depends on multiple previous terms in recurrence sequence

Banach fixed-point theorem http://en.wikipedia.org/wiki/Banach_fixed-point_theorem I'm wondering if the theorem still applies if the recurrence relation is, for example, ...
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2answers
44 views

Accumulation points and closed sets

Denote by $F$ the set of all accumulation points of $(x_{n})$. We define an accumulation point $x \in \mathbb{R}$ if there exists a subsequence $(x_{n_{k}})$ of $(x_{n})$ (being the latter a bounded ...
0
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1answer
28 views

Subsequence converging to inf of sup

Let $(x_n)$ be a bounded sequence, and for each $n\in\mathbb{N}$ let $s_n=\sup\{x_k:k\geq n\}$ and $S=\inf\{s_n\}$. I need to show that there exists a subsequence of $(x_n)$ that converges to $S$. I ...
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0answers
24 views

Question about a proof in Lang's undergraduate analysis

This is from page 580 of Lang's undergraduate analysis (2nd edition). I have difficulty in understanding the proof, hope that someone here can enlighten me. My questions are: i) On line 5 of the ...