1
vote
2answers
87 views

Questions about Proof of Lusin's Theorem

I am reviewing my analysis notes, and having trouble understanding certain parts of the proof to Lusin's theorem. $\textbf{Lusin's Theorem}$: Let $F: [0,1] \rightarrow [0,\infty)$ be a nonnegative, ...
0
votes
2answers
37 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
3
votes
1answer
63 views

$f$ is differentiable. If $\lim_{x \to c}f'(x)$ exists, then this limit must be $f'(c)$.

Please prove: Let $f:(a,b) \to R$ be differentiable function, and let $c \in (a,b). $ If $\lim_{x \to c}f'(x)$ exists and is finite, then this limit must be $f'(c)$. I tried doing it directly but ...
3
votes
3answers
39 views

Proof that for every non empty $A \subset \mathbb{N}$ there's a $f:\mathbb{N} \rightarrow \mathbb{N}$ with $f \circ f =f$ and $f(N)=A$

I need to prove that, for every non empty $A \subset \mathbb{N}$, there's a $f: \mathbb{N} \rightarrow \mathbb{N}$ with $f \circ f =f$ and $f(\mathbb{N})=A$ Unfortunately I have no idea where to ...
0
votes
1answer
43 views

Help with proof for solving an ODE using contraction mapping theorem

I'm trying to follow a proof for solving the ODE $$\frac{df}{dx} = (f(x)+x)x$$ For $0 \leq x \leq 1$ with the initial condition $ f(0)=0$. The proof I am following goes like this Define ...
2
votes
2answers
31 views

Proving a theorem about the limit of a function

The theorem is as follows: $$\exists L\in\mathbb R\ \bigg(\lim_{x \to c}f(x)=L\bigg)\iff\forall\varepsilon>0\ \exists\delta>0\ \forall x_1,x_2\in B_{\delta}(c):|f(x_1)-f(x_2)|<\varepsilon$$ ...
2
votes
0answers
75 views

Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let ...
1
vote
1answer
40 views

Uncountability of a nonmeasurable set

As per the Vitali's theorem, every measurable set of positive measure has a subset which is nonmeasurable. Which proceeds by defining a rational equivalence, followed by using the axiom of choice on ...
0
votes
2answers
32 views

How to quantify this statement

How do I quantify this statement? Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. When I am trying to prove this, I am led (in the course of ...
1
vote
1answer
40 views

Is there a more concise way?

I am currently trying to prove Taylor's Theorem, this is the proof I am given: I am finding this very long and hard to follow, can anyone give me a more concise, clear proof? Thanks
1
vote
1answer
98 views

Proving the chain rule by first principles

I'm currently trying to prove: $(f(g))'(a)=f'(g(a))*g'(a)$ I have been given a proof which manipulates: $f(a+h)=f(a)+f'(a)h+O(h)$ where $O(h)$ is the error function. However, I would like to have a ...
0
votes
1answer
44 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 ≤ ...
2
votes
1answer
157 views

Proof concerning equivalent definition of supremum using limits

I believe I have a proof to the following theorem: Let $S$ be a subset of $\mathbb{R}$ that is non-empty and bounded above. $s \in \mathbb{R}$ is the supremum iff $s$ is an upper bound of $S$ and for ...
0
votes
0answers
29 views

Need help on analytically proving the monotonicity of an inexplicit integral

I have the following function which I have numerically investigated to be monotonically increasing in $\nu$: $D(x;\mu,\nu) =\frac{1}{\sqrt{2\pi \cdot \nu}}\int_{-\infty}^{\infty}e^{-\theta \cdot x} ...
0
votes
1answer
29 views

Analytic, never zero on domain D graph and its local max. min

If $f$ is analytic and never zero on domain D, then $|f(z)|$ has no local minimum or maximum. Now, If I use the fact that $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then I can show ...
0
votes
0answers
31 views

Prove that $s \not= \emptyset $ by showing that at least one of $|a|$ or $|b|$ is an element of $S$.

. I'm trying to show an alternative proof to Bezout's Lemma (let $a, b \in \mathbb{Z}$, then there exists $x,y \in \mathbb{Z}$ such that $gcd(a, b) = ax + by$). Heres one of the steps in proving it: ...
1
vote
1answer
67 views

Possible book correction or am I missing something?

Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
1
vote
2answers
65 views

$\forall x \in \mathbb{R}$ show that $x=\sum_{n=1}^\infty k_na_n = \prod_{n=1}^{\infty}m_na_n$ …

Yet again, another cool problem from the book "problems in mathematical analysis" by Piotr & Witkowski: Prove that if $a_n \neq 0$, $n=1,2,\cdots$ and $\displaystyle \lim_{n \to \infty} a_n = 0$, ...
0
votes
1answer
86 views

If the partial derivatives are continuous then the function is differentiable in the context of 3 dimensions

Context: This question has been bugging me for a while, mainly due to no knowledge of linear algebra and availability of only an ugly book Stewart's Calculus. There is a sufficient condition for a ...
1
vote
2answers
119 views

Linear surjective isometry then unitary

Basically what I'm trying to show is $\forall h_1, \ h_2 \in \mathscr{H}$ and $U: \mathscr{H} \rightarrow \mathscr{K}$ then $\langle Uh_1, \ Uh_2\rangle_\mathscr{K} = \langle h_1, \ h_2 ...
1
vote
1answer
85 views

Tangent Cone is a cone?

I first give two definitions. Def1: A set $S$ is a cone if $x \in S, \lambda \geq 0 \implies \lambda x \in S$. Def2: Let $S$ be any set (we may assume $\mathbb{R}^n$ with the usual Euclidean ...
3
votes
2answers
210 views

Stuck with a tricky existence proof

Show that there exists a continuous function $f: [-1, 1] \rightarrow \mathbb{R}$ such $f(0) = 1$ and $f(x) = \frac{2-x^2}{2} \cdot f(\frac{x^2}{2-x^2})$ $\forall x \in [-1, 1]$ I tried putting ...
1
vote
1answer
55 views

Question about convergence of a power series and when the series is not zero

Following is a past exam question I am trying to solve as a preparation for my own exam. Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers with $a_n \leq M$ for some $M\in\mathbb{R}^{+}$ and ...
1
vote
3answers
75 views

Show that $(x_n)$ is decreasing and find its limit.

Let $0<x_1<1$. For $n \in \mathbb{N}$, let $x_{n+1}=1- \sqrt{1-x_n}$. Show that $(x_n)$ is decreasing and find its limit. I did: $$x_{n+1} = 1- \sqrt{1-x_n}$$ $$x_{n+1} - x_n= 1- \sqrt{1-x_n} - ...
1
vote
1answer
58 views

Do this algorithm terminates?

Let $x \in \mathbb{R}^p$ denote a $p$ dimensional data point (a vector). I have two sets $A = \{x_1, .., x_n\}$ and $B = \{x_{n+1}, .., x_{n+m}\}$. So $|A| = n$, and $|B| = m$. Given $k \in ...
1
vote
2answers
36 views

Limit and maximum: IVT

Let $f$ be a function defined and continuous on $\mathbb{R}$. Assume that $f(a) > 0$ for some $a \in \mathbb{R}$ and that $$\lim_{x\to \infty} f(x) = 0 = \lim_{x\to -\infty}f(x)$$ Show that ...
1
vote
2answers
196 views

Limit of functions and the Binomial Theorem

If $n \geq 2$ is an integer, show $n^{1/n} = 1 + h$; where $h \leq \sqrt{ \dfrac{2}{n-1}}$ Then Deduce that: $\lim\limits_{n \to \infty} n^{1/n} = 1$ Hint: Since $n>1$, $n^{1/n}>1$. So, ...
0
votes
1answer
68 views

Let $\delta$ be a linear functional equipped with the sup-norm. Show that $\delta$ is bounded and compute its norm.

Let $\delta:C([0,1])\rightarrow\mathbb{R}$ be the linear functional at the origin: $\delta(f) = f(0)$. If $C([0,1])$ is equipped with the sup-norm $$\|f\|_{\infty} = \sup_{0\leq x\leq 1}|f(x)|.$$ Show ...
1
vote
1answer
32 views

What is the Density Theorem in this context?

I have this exercise: Define $K:C([0,1])\rightarrow C([0,1])$ by $$Kf(x) = \int_0^1 k(x,y)f(y)dy,$$ where $k:[0,1]\times [0,1]\rightarrow \mathbb{R}$ is continuous. Prove that $K$ is bounded and ...
0
votes
1answer
40 views

Showing that a set $D$ is closed and open

I have this theorem in my book: Suppose that $f$ is continuous in the rectangle $$R = \{(t,u) \mid |t-t_0|\leq T, |u-u_0|\leq L\} $$ and that $$|f(t,u)|\leq M \text{ if } (t,u)\in R$$ Let $\delta = ...
2
votes
1answer
55 views

Help with understanding a proof on Ordinary Differential Equations

I have this theorem in my book: Suppose that $f$ is continuous in the rectangle $$R = \{(t,u) | |t-t_0|\leq T, |u(t)-u_0|\leq L\} $$ and that $$|f(t,u)|\leq M \text{ if } (t,u)\in R$$ Let $\delta = ...
0
votes
3answers
84 views

Question on Proof of the Contraction Mapping Theorem

Contraction Mapping Theorem If $T\colon X\to X$ is a contraction mapping on a complete metric space $(X,d)$ then there is exactly one solution $x\in X$. Proof: Let $x_0$ be any point in $X$. We ...
1
vote
1answer
74 views

Prove Heine-Borel Thm

Prove Heine-Borel Theorem: "A subset $S$ of $\mathbb{R}$ is compact if and only if every open cover for $S$ has a finite subcover." Suggestions: Let $S \subset \mathbb{R}$. If every open cover for ...
2
votes
1answer
148 views

Open Cover for a Compact Subset

I am doing some extra exercises for an Analysis class, and I found this one. We haven't seen much of what an open cover is, but I want to learn it. So, here it goes, and thank you everyone! Let ...
0
votes
1answer
28 views

Prove that $Tu(x)$ is a contraction. $Tu(x) = -\lambda\int_0^1g(x,y)\sin(u(y))\,dy$

I want to show that $Tu(x)$ is a contraction where $$Tu(x) = -\lambda\int_0^1g(x,y)\sin(u(y))\,dy$$ and $$g(x,y) = \begin{cases} x(1-y) & 0\leq x\leq y\leq 1, \\ y(1-x) & 0\leq y \leq x \leq ...
1
vote
1answer
228 views

Open cover with no finite subcover

Let (x_n) be a sequence, let $L$ ∈ R, and for each ϵ>0, {k ∈ N : x_k ∈ B($L$; ϵ)} Suppose S is not a compact subset of R. There is some ϵ_L > 0, such that {k ∈ N : x_k ∈ B($L$; ϵ_L)} is finite. ...
0
votes
1answer
58 views

Convergent subsequence

1) Let (x_n) be a sequence and let L ∈ R. Suppose that for each ϵ > 0, {k ∈ N : x_k ∈ B(L; ϵ)} is infinite. Show that (x_n) has a subsequence converging to L. ...
0
votes
1answer
29 views

Need help clarifying a proof ( limSn=SupS)

Let $S$ be a bounded nonempty subset of $R$ such that $Sup(S)$ is not in $S$. Prove $\exists$ a sequence $(S_n)$ of points that belong to $S$ such that $ limS_n=Sup(S)$. Let $t=Sup(S)$.then for ...
4
votes
2answers
125 views

If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. Question on proof.

Let $X,Y$ be metric spaces and $f:X\rightarrow Y$. If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. The text I am using proves this proposition like so: Suppose ...
0
votes
1answer
77 views

Question about intervals and infima/suprema

Let $L(E)$ be the set of lower bounds of a set $E$ and $(S, \le)$ a partially ordered set. For each $s \in S$, let $$ \langle s] := \{x \in S \mid x \le s\} $$ and $$ [s\rangle := \{x \in S \mid ...
2
votes
1answer
111 views

Problem of proofs

I've been away from math for a long time ,and while I was trying to relearn it using Courant and Fritz 's booknon calculus,I loved the explanations but I couldn't solve any exercices(they're almost ...
0
votes
1answer
246 views

Proof of the continuous function having tangent plane has directional derivatives

Suppose that the continuous function $f: \Bbb R^2 \to \Bbb R$ has a tangent plane at the point $(x_0, y_0, f(x_0, y_0))$ Prove that the function $f$ has directional derivatives in all directions at ...
0
votes
1answer
86 views

Correctness of Analysis argument with Cauchy sequences

Let $(x_n)$ and $(y_n)$ be Cauchy sequences in $(X, d)$. Show that $(x_n)$ and $(y_n)$ converge to the same limit iff $d(x_n, y_n) \rightarrow 0$ Proof $\rightarrow$ Suppose $(x_n) \to a$ and $(y_n) ...
4
votes
2answers
2k views

A continuous function $f$ from a closed bounded interval $[a, b]$ into $\mathbb{R}$ is uniformly continuous

I am reading (as a supplement) the book Basic Real Analysis, by Anthony Knapp. Before I proceed into reading a proof, I want to be sure that the result seems obvious. Yet, I am having trouble seeing ...
2
votes
1answer
263 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 ...
1
vote
3answers
679 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 ...
0
votes
1answer
100 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 ...
3
votes
2answers
217 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 ...
4
votes
4answers
209 views

Proof of the irrationality of $\sqrt{3}$ - logic question

Prove $\sqrt{3}$ is irrational. (Proof by contradiction). Let $\sqrt{3}$ be a rational number in simplest form $\frac pq$. So squaring both sides of $\sqrt{3}=\frac pq$ we get $3=(\frac {p}{q})^2$ ...
0
votes
1answer
103 views

Proof for Convergent Sequences [duplicate]

Possible Duplicate: Did I underestimate the limit proof? Let $(a)_{n\in \Bbb N}$ and $(b)_{n\in \Bbb N}$ be sequences of real numbers such that $a_n$ $\le$ $b_n$ for all $n\in \Bbb N$. ...