0
votes
1answer
47 views

Real Analysis Question: derivatives

Let $$f''(x)+p(x)\cdot f(x)=0$$ and $$g''(x)+p(x)\cdot g(x)=0$$ where $a<x<b$. 1 ) Show that $W=f'g-fg'$ is a constant on $(a,b)$. 2 ) Prove: If W$\neq$0 and $f(x_1)=f(x_2)=0$ where $a \lt x_1 ...
2
votes
4answers
265 views

How to deduce the following trig relation?

How can I deduce: $$\sqrt{|x|}\sin(\frac{1}{x}) \le \sqrt{|x|}$$?? I know of the relation. $$\sin(u) \le u$$ $$u = \frac{1}{x}$$ $$\sin(1/x) \le \frac{1}{x}$$ But nothing related to $\sqrt{x}$ ...
0
votes
0answers
42 views

Correctness of proof based on nested interval theorem

I am given that for positive integer n, let $a_n=1-(1/n)$ and $b_n=1+(1/n)$. $I_n=[a_n,b_n]$. I am to show that 1) the hypothesis of the nested interval theorem are satisfied, 2) find the point of ...
1
vote
1answer
102 views

Proof that arithmetic and geometric mean converge

I need some help with understanding a part of this proof and also writing it up correctly. Given $a_n\geq a_{n+1}\geq b_{n+1} \geq b_n$ with $a_1=a$ and $b_1=b$. I am also given that ...
1
vote
3answers
137 views

Can the proof of Theorem 1.20 (b) in the book, The Principles of Mathematical Analysis by Walter Rudin, 3rd ed., be improved?

I'm reading Walter Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS, third edition, and am at Theorem 1.20(b), where he states and proves that between any to real numbers, there is a rational; that is, if ...
1
vote
2answers
142 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
38 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
65 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
41 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
47 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
35 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
88 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
41 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
35 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
41 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
177 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
49 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
252 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
31 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
32 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
70 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
68 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
104 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
161 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
92 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
216 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
77 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
38 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
258 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
76 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
35 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
56 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
90 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
77 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
153 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
30 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
288 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
60 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
31 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
128 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
82 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
115 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
276 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
87 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
295 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 ...