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...
0
votes
1answer
48 views
Show $C\geq \mathrm{max}\left \{ A,B \right \}$.
Let $\sum_{n=0}^{\infty}a_{n}x^{n}$ and $\sum_{n=0}^{\infty}b_{n}x^{n}$ be the power series with the convergent of radius respectively $A>0$ and $B>0$.
Define $c_{n}=\mathrm{min}\left \{ ...
0
votes
0answers
43 views
Uniform convergence of $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$.
Given that the series $\sum_{n=2}^{\infty}\dfrac{\mathrm{ln}(n)^{p}}{n}$ is convergent iff $p<-1$.
Show that $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$ is uniform convergent as $x \in ...
0
votes
1answer
41 views
Find for all value of constant $a>0$; the interval of convergence of the power series $\sum_{n=0}^{\infty}\frac{1}{1+a^{n}}x^{n}$
Find for all value of constant $a>0$; the interval of convergence of the power series
$\sum_{n=0}^{\infty}\frac{1}{1+a^{n}}x^{n}$.
What I have tried is; if we let $b_{n}=\frac{1}{1+a^{n}}x^{n}$ so ...
1
vote
2answers
63 views
If $K$ is compact, then $C(K,\mathbb{R}^n)$ is a Banach space under the norm $\|f\|=\sup_{x\in K} \|f(x)\|$
Let $K$ be a topological space that is compact. Show that the space $C(K,\mathbb{R}^n)$ of all the continuous functions $f:K\to\mathbb{R}^n$ is a Banach space with the norm $\|f\|=\sup_{x\in K} ...
1
vote
4answers
39 views
Is $(a_n)_n$ with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ a Cauchy sequence?
Let $0 < q < 1$ and $(a_n)_n$ be a squence with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ for all $n ∈ ℕ$.
I need to show that this is a Cauchy sequence. I'm not sure how to start this one, as we ...
2
votes
0answers
26 views
Growth of partial sums of a divergent series
I am trying to find how the product $\prod_{n=0}^Me_n$ grows with $M$, when $$e_n=1+\frac{a_1}{t+n}+\frac{a_2}{(t+n)^2}+\dots$$
with large $t$. So as a first estimate I took $e_n=1+\frac{a_1}{t+n}$ so
...
0
votes
1answer
33 views
Show convergence for this sequence only by using the definition
I need to prove convergence for
$(b_n)_{n ∈ ℕ}=\left(\frac{(-1)^nn}{2n+1}\right)_{n∈ℕ}$ and also show the limit.
I may only use the following definition: $∀ɛ > 0∃n_0∈ℕ∀n≥n_0:|a_n-a|< ɛ$.
So far ...
11
votes
1answer
141 views
Uniform convergence of $\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$
I'm not sure wether or not the following sum uniformly converge on $\mathbb{R}$ :
$$\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$$
Can someone help me with it? (I can't use Dirichlet' ...
0
votes
1answer
18 views
Show relation for integrals
Let $f \in C^{1}([a,b];\mathbb{R})$ and $|f'(x)-f'(y)| \le L |x-y|$
then we have $|\int_a^b f(x) dx -f(\frac{a+b}{2})(b-a)| \le L\frac{(b-a)^3}{4}$.
I have troubles to show this inequality. the ...
1
vote
1answer
51 views
A question about “nice” functions
Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us call such functions ...
1
vote
2answers
63 views
Proving that Euclidean space having the infinity metric is a complete metric space (stuck)
I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space.
I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ ...
0
votes
1answer
98 views
Changing order of derivatives
I would like to rewrite the following expression
$$\frac{d^i}{dx^i}\left\{f(x)\left[\frac{d^jf(x)}{dx^j}\right]\left[\frac{d^kf(x)}{dx^k}\right]\right\}$$
into the form
$$D f(x)^3,$$
with $D$ ...
3
votes
3answers
39 views
Prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ is a metric on $\mathbb{R}^k$
I am trying to prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,...,k\}$ is a metric on $\mathbb{R}^k$. So far I know that $\text{dist}\left({x,y}\right) \leq ...
10
votes
2answers
194 views
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
1
vote
0answers
53 views
Let$ f : [0, 1]^2 \to R$ such that $f(x, y)$ is continuous in $x$ for each fixed $y$ and conversely also. Is $f $ continuous?
Let$ f : [0, 1]^2 \to R$ such that $f(x, y)$ is continuous in $x$ for each fixed $y$ and continuous in $y$ for each fixed x. Does it follow that $f$ is continuous?
0
votes
1answer
32 views
The sum of the integration of g and $g^{-1}$
Let $g$ be a strictly increasing continuous function mapping $[a,b]$ onto
$[A,B]$, and, as usual, let $g^{-1}: [A,B] \to [a,b]$ denote its inverse function.
Use geometric insight to visualize the ...
1
vote
1answer
51 views
Integral over null set is zero but integral of Dirac delta function is 1
We know integral of any function over a null set is zero.
But for Dirac delta function ($\delta=+\infty$ iff $x=0$ otherwise $\delta=0$)
$$
\int_{-\infty}^{+\infty}\delta =\int_0^0\delta =1.
$$
Is it ...
5
votes
1answer
85 views
Continuous function differentiable on $[0,1]\setminus\mathbb{Q}$, but nondifferentiable on all of $\mathbb{Q}\cap[0,1]$?
I'm trying to work out an example of a continuous function which is differentiable at all irrationals but nondifferentiable at all rationals in $[0,1]$.
Since $\mathbb{Q}$ is countable, list it as ...
4
votes
2answers
52 views
If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$
Let $f\colon\mathbb R\to\mathbb R$ be twice differentiable with $f(x)\to 0$ as $x\to\infty$ and $f''$ bounded.
Show that $f'(x)\to0$ as $x\to\infty$.
(This is inspired by a comment/answer to a ...
5
votes
2answers
62 views
Bilinear forms on C[0,1]
Let $C[0,1]$ be the vector space of real-valued continuous functions on $[0,1]$. Then
$$B(f,g) = \int_0^1{f(x)g(x)\, dx}$$
is a bilinear form on $C[0,1]$. More generally, if $k:[0,1]^2\rightarrow ...
1
vote
1answer
35 views
Understanding when convergence implies uniform convergence for sequences of non-continuous functions
I am working on the following problem:
Let $(f_n)$ be a sequence of functions $[a,b] \rightarrow \mathbb{R}$ such that: (i) $f_n(x)≤0$ if $n$ is even, $f_n(x)≥0$ if $n$ is odd; (ii) ...
12
votes
3answers
137 views
Which real functions have their higher derivatives tending pointwise to zero?
Let $\mathrm C^\infty\!(\Bbb R)$ be the space of infinitely differentiable functions $f:\Bbb R\rightarrow\Bbb R$, and define the subspace$$A:=\{f\in\mathrm C^\infty\!(\Bbb R):(\forall x\in \Bbb ...
1
vote
0answers
36 views
Product of Fourier integrals
I am interested in solving the following integral:
\begin{equation}
I =\int dx_{3}\psi^{\star}(x_{3})\int dx_{1}\psi(x_{1})\int dq_{1}X(q_{1})e^{iq_{1}(x_{3}-x_{1})}\int dx_{2}\psi(x_{2})\int ...
0
votes
1answer
52 views
Liouville's formula
I have some questions concerning a proof of Liouville's formula:
$$W'(t)=\text{tr}(A) W(t)$$ where $W$ is the Wronskian of the homogenous ODE.
If the vectors in the columns of the fundamental matrix ...
2
votes
2answers
66 views
When $\int |f|=\left|\int f\right|$ holds?
I was just wondering when did the equality hold for the following inequality:
$$\left|\int_{R^d}f(x)\, d x\right|\leq\int_{R^d}|f(x)|\, d x$$
where $f:R^d\to R$ is Lebesgue integrable on $R^d$.
...
1
vote
0answers
33 views
What's the exact definition for convolution?
I tried to solve the problem in Stein's Real analysis, 1ed, P94, Ex 21 (c), which asked to show that for any two measurable functions $f,g$ on $R^d$, the convolution of $f$ and $g$, $$(f\ast ...
1
vote
0answers
28 views
What's the need of $^{S}_{T}$ in $f^{S}_{T}:S\rightarrow T$?
I'm reading Lang's Undergraduate Analysis:
In the chapter about mappings, he says that we should denote the set of arrival and the set of departure with the following notation:
...
0
votes
2answers
49 views
If $E = \{ x \in \mathbb{R}: \sin(\frac1{x}) = 1\}$ then $l = 0$ is a limit point of E
If $E = \{ x \in \mathbb{R}: \sin\left(\frac{1}{x}\right) = 1\}$, then $l = 0$ is a limit point of $E$.
I have a proof here but I don't quite understand a few points, I hope someone can explain it a ...
0
votes
0answers
29 views
analysis: limit of product of sequences [duplicate]
I would really appreciate help with this question:
Show that if the limit $\lim_{n \to \infty} {a_n} = 0$ and sequence $\{b_n\}$ is bounded, then $\lim_{n\to \infty} a_nb_n =0$
thanks
0
votes
1answer
31 views
what does “in wide sense” mean?
I came across the statement "the sequence increases(in wide sense)".
So my doubt is what does author mean by wide sense?I came across this in number theory book
3
votes
1answer
38 views
Differentiability implies Lipschitz continuity
Let $f:[0,1]\to\mathbb{R}$ be a continuous function and suppose $f$ is differentiable at $x_0\in [0,1]$. Is it true that there exists $L>0$ such that $\lvert f(x)-f(x_0)\lvert\leq L\lvert ...
1
vote
2answers
236 views
A less known definition of the definite integral of a continuous function
The definite integral of a continuous function can be defined using the bounded monotone sequence property: see Osgood's Functions of Real Variables, p.110.
(link to full book) (screenshots: page ...
1
vote
2answers
58 views
Show uniform convergence of $\sum\limits_n(1-x^2)^2 x^n$ on $[0,1]$
I'm trying to show that if $u_n(x) = (1-x^2)^2 x^n$ then $\sum u_n$ is uniformly convergent on $[0,1]$.
The only thing I can think to use is Weierstraß M-test to show that $u_n < M_n$ where $\sum ...
2
votes
1answer
36 views
Is it possible to write any bounded continuous function as a uniform limit of smooth functions
Is $C^\infty(\mathbb{R})\subset C_b(\mathbb{R})$ dense? I.e. is any continuous bounded function $f:\mathbb{R}\to\mathbb{R}$ the uniform limit of smooth functions?
On any bounded interval this is ...
3
votes
1answer
69 views
Most elegant/simple proof of the irrationality of $\pi$
What is the most elegant/shortest proof of this? The proofs I have seen are quite long, unlike the proof of the irrationality of $e$.
thanks
7
votes
5answers
123 views
To show that function is constant
Let $f$ be defined on $\mathbb{R}$ and suppose that |$f(x)$ - $f(y)$| $\leq$ $(x-y)^2$ $x,y \in\mathbb{R}$. Here I have to show that $f$ is a constant function.
I think I have to show that $f'(x)$ = ...
3
votes
2answers
39 views
Rudin: Real & Complex Analysis Thm 1.10
$\textbf{Theorem:}$ If $\mathcal{F}$ is any collection of subsets of $X$, there exists a smallest $\sigma$-algebra $\mathcal{M}^{*}$ in $X$ such that $\mathcal{F} \subset \mathcal{M}^{*}$.
...
10
votes
7answers
468 views
Does $\,x>0\,$ hint that $\,x\in\mathbb R\,$?
Does $x>0$ suggest that $x\in\mathbb R$?
For numbers not in $\,\mathbb R\,$ (e.g. in $\mathbb C\setminus \mathbb R$), their sizes can't be compared.
So can I omit "$\,x\in\mathbb R\,$" and ...
0
votes
0answers
40 views
Riemann integration show if f is integrable then g is integrable
I have the following question asked of me.
Suppose that $f$ and $g$ are bounded functions on $[a,b]$
and there exists a point $c\in[a,b]$ such that $f(x) = g(x)$ for every $x\neq c$.
Prove that $U(f) ...
4
votes
1answer
67 views
$\iint f(x,y)\,dxdy$ and $\iint f(x,y)\,dydx$ exist but $f$ not integrable on $[0,1]\times[0,1]$
I want to look for a function $f(x,y)$, whose support is inside $[0,1]\times[0,1]$, such that $\int_0^1\!\int_0^1\!f(x,y)\,dxdy$ and $\int_0^1\!\int_0^1\!f(x,y)\,dydx$ both exist, but $f(x,y)$ is not ...
9
votes
3answers
158 views
Dirac Delta or Dirac delta function?
Is Dirac delta a function? What is its contribution to analysis?
What I know about it:
It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
1
vote
3answers
46 views
If $x \leq g(x) \leq x^2-x+1$ where $x \in [0,2]$, can we say that $g(x)$ is continuous at $x=1$?
If $x \leq g(x) \leq x^2-x+1$ where $x \in [0,2]$, can we say that $g(x)$ continuous at $x=1$ ?
Is $g(x)$ continuous in $[0,2]$?
1
vote
0answers
25 views
Is the Cauchy principal value “invariant” under change of variables?
Let $f \in C^{\gamma}_c(\mathbb{R}) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties:
1) K smooth everywhere except ...
1
vote
1answer
53 views
Flow of $rot \overrightarrow{F}$
We've got vector field: $\overrightarrow{F} = \begin{bmatrix} yz\\x^3z\\e^z\end{bmatrix}$. I want to compute flow of $rot\overrightarrow{F} $($=curl \overrightarrow{F}$) through the area of the side ...
1
vote
1answer
29 views
Proving integrability in integration by parts in Rudin's text
Integration by parts, as stated in W. Rudin's Principles of Mathematical Analysis, Theorem 6.22, goes as follows:
Suppose F and G are differentiable functions in $[a,b]$, $F'=f\in \mathcal{R}$, ...
1
vote
1answer
26 views
sum of monotonic increasing and monotonic decreasing functions
I have a question regarding sum of monotinic increasing and decreasing functions. Would appreciate very much any help/direction:
Consider an interval $x \in [x_0,x_1]$. Assume there are two functions ...
0
votes
1answer
97 views
Which topics of real-analysis should be studied if you have already done calculus
Which parts of real-analysis are worth studying if you have already taken several calculus courses? I know that real-analysis is more 'rigurous', but still I wonder whether it is worth to again go ...
2
votes
1answer
49 views
Is the inverse function smooth?
Imagine that we have a function $Inv$ that maps $A \rightarrow A^{-1}$, where A is an invertible square matrix. now my questions is: how do i see that this function is arbitrarily often ...
4
votes
0answers
44 views
Evaluate $\int_0^{\infty} \frac{1-e^{-ax}}{x e^x} dx$
I found two different approaches, both is giving the same answer.
Fubini:
$$
\begin{align}
\int_0^{\infty} \frac{1-e^{-ax}}{x e^x} \,dx &= \int_0^{\infty} e^{-x} \int_0^a e^{-xy} \,dy\, dx \\
...
1
vote
1answer
25 views
$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$
For $f\in L^{p}$, $p \in [1,\infty)$
we want to prove:
$$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$
I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
