1. ### I ##(a_n) ## has +10,-10 as partial limits. Then 0 is also a partial limit

Problem: If sequence ## (a_n) ## has ##10-10## as partial limits and in addition ##\forall n \in \mathbb{N}.|a_{n+1} − a_{n} |≤ \frac{1}{n} ##, then 0 is a partial limit of ## (a_n) ##. Proof : Suppose that ## 0 ## isn't a partial limit of ## (a_n) ##. Then there exists ## \epsilon_0 > 0 ## and...
2. ### Can I use recursion/induction to show that N <= x < N+1 for x real

Homework Statement:: Show that for every real number ##x## there is exactly one integer ##N## such that ##N \leq x < N+1##. (This integer is called the integer part of ##x##, and is sometimes denoted ##N = \lfloor x\rfloor##.) Relevant Equations:: N/A I have tried reading the solution given...
3. ### Is my proof that multiplication is well-defined for reals correct?

I have referred to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ to check my answer. The way I thought of the problem: I know ##xy = \mathrm{LIM}_{n\to\infty} a_n b_n## and I know ##x'y = \mathrm{LIM}_{n\to\infty} a'_n b_n##. Thus if ##xy=x'y##, maybe I can try showing...
4. ### Purpose of \varepsilon' := \min\left(\frac{\varepsilon}{3M_2}, \frac\varepsilon 3\right)

I refer to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ I am having trouble understanding the purpose / motivation behind using the min as in ##\delta := \min\left(\frac{\varepsilon}{3M_1}, 1\right)## and ##\varepsilon' := \min\left(\frac{\varepsilon}{3M_2}...

In Tao's Analysis 1, Lemma 5.3.6, he claims that "We know that ##(a_n)_{n=1}^{\infty}## is eventually ##\delta##-steady for everyvalue of ##\delta>0##. This implies that it is not only ##\epsilon##-steady, ##\forall\epsilon>0##, but also ##\epsilon/ 2##-steady." My question is, why do we need...
6. ### Proof that two equivalent sequences are both Cauchy sequences

Let us just lay down some definitions. Both sequences are equivalent iff for each ##\epsilon>0## , there exists an N>0 such that for all n>N, ##|a_n-b_n|<\epsilon##. A sequence is a Cauchy sequence iff ##\forall\epsilon>0:(\exists N>0: (\forall j,k>N:|a_j-a_k|>\epsilon))##. We proceeded by...
7. ### I Will ##M_i = m_i## if an interval is made vanishingly small?

We define : $$M_i = sup \{f(x) : x \in [x_{i-1}, x_i ] \}$$ $$m_i = inf \{f(x) : x \in [ x_{i-1}, x_i ] \}$$ Now, if we make the length of the interval ##[x_{i-1}, x_i]## vanishingly small, then would we have ##M_i = m_i##? I have reasons for believing so because as the size of the interval is...
8. ### I How to prove that ##f## is integrable given that ##g## is integrable?

We have a function ##f: [a,b] \mapsto \mathbb R## (correct me if I'm wrong but the range ##\mathbb R## implies that ##f## is bounded). We have a partition ##P= \{x_0, x_1 , x_2 \cdots x_n \}## such that for any open interval ##(x_{i-1}, x_i)## we have $$f(x) =g(x)$$ (##g:[a,b] \mapsto \mathbb...

19. ### Single Point Continuity - Spivak Ch.6 Q5

Hey Guys, I posed this on Math Stackexchange but no one is offering a good answering. I though you guys might be able to help :) https://math.stackexchange.com/questions/3049661/single-point-continuity-spivak-ch-6-q5
20. ### Proof about limit superior

Homework Statement 2. Relevant equation Below is the definition of the limit superior The Attempt at a Solution I tried to start by considering two cases, case 1 in which the sequence does not converge and case 2 in which the sequence converges and got stuck with the second case. I know...
21. ### Prove that there exists a graph with these points such that...

Homework Statement Let us have ##n \geq 3## points in a square whose side length is ##1##. Prove that there exists a graph with these points such that ##G## is connected, and $$\sum_{\{v_i,v_j\} \in E(G)}{|v_i - v_j|} \leq 10\sqrt{n}$$ Prove also the ##10## in the inequality can't be replaced...
22. ### Curve and admissible change of variable

Homework Statement If I have the two curves ##\phi (t) = ( \cos t , \sin t ) ## with ## t \in [0, 2\pi]## ##\psi(s) = ( \sin 2s , \cos 2s ) ## with ## s \in [\frac{\pi}{4} , \frac{5 \pi}{4} ] ## My textbook says that they are equivalent because ##\psi(s) = \phi \circ g^{-1}(s) ## where ##...
23. ### Need help formalizing "T is an open set"

Homework Statement Let ##S\subseteq \Bbb{R}## and ##T = \{ t\in \Bbb{R} : \exists s\in S, \vert t-s\vert \lt \epsilon\}## where ##\epsilon## is fixed. I need to show T is an open set. Homework Equations n/a The Attempt at a Solution Let ##x \in T##, then ##\exists \sigma \in S## such that ##x...
24. ### Image of a f with a local minima at all points is countable.

Homework Statement Let ##f:\Bbb{R} \to \Bbb{R}## be a function such that ##f## has a local minimum for all ##x \in \Bbb{R}## (This means that for each ##x \in \Bbb{R}## there is an ##\epsilon \gt 0## where if ##\vert x-t\vert \lt \epsilon## then ##f(x) \leq f(t)##.). Then the image of ##f## is...
25. ### I Learning the theory of the n-dimensional Riemann integral

I would like to learn (self-study) the theory behind the n-dimensional Riemann integral (multiple Riemann integrals, not Lebesgue integral). I am from Croatia and found lecture notes which Croatian students use but they are not suitable for self-study. The notes seem to be based on the book: J...
26. M

### I Two questions about derivatives

In Rudin, the derivative of a function ##f: [a,b] \to \mathbb{R}## is defined as: Let ##f## be defined (and real-valued) on ##[a,b]##. For any ##x \in [a,b]##, form the quotient ##\phi(t) = \frac{f(t) - f(x)}{t-x}\quad (a < t <b, t \neq x)## and define ##f'(x) = \lim_{t \to x} \phi(t)##, if the...
27. ### I Question regarding a sequence proof from a book

I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null...
28. ### Show that ##\frac{1}{x^2}## is not uniformly continuous on (0,∞).

Homework Statement Show that ##f(x)=\frac{1}{x^2}## is not uniformly continuous at ##(0,\infty)##. Homework Equations N/A The Attempt at a Solution Given ##\epsilon=1##. We want to show that we can compute for ##x## and ##y## such that ##\vert x-y\vert\lt\delta## and at the same time ##\vert...
29. ### Distance of a point from a compact set in ##\Bbb{R}##

Homework Statement Let ##K\neq\emptyset## be a compact set in ##\Bbb{R}## and let ##c\in\Bbb{R}##. Then ##\exists a\in K## such that ##\vert c-a\vert=\inf\{\vert c-x\vert : x\in K\}##. 2. Relevant results Any set ##K## is compact in ##\Bbb{R}## if and only if every sequence in ##K## has a...
30. ### Showing that an exponentiation is continuous -- Help please...

Homework Statement Let ##p\in\Bbb{R}##. Then the function ##f:(0,\infty)\rightarrow \Bbb{R}## defined by ##f(x):=x^p##. Then ##f## is continuous. I need someone to check what I've done so far and I really need help finishing the last part. I am clueless as to how to show continuity for...