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, and integration through the fundamental theorem of calculus.

learn more… | top users | synonyms

2
votes
2answers
46 views

Prove that there exists $x_0\in [a,b]$ such that $ \sum_{i=1}^{n} k_i \int_{x_0}^{x_i} fdt=0$

Let $f$ is a continuous function on $[a,b]$, $x_1,x_2,\ldots,x_n\in [a,b]$, $k_1,k_2,\ldots,k_n>0$. Prove that there exists $x_0\in [a,b]$ such that $$k_1\displaystyle \int_{x_0}^{x_1} ...
3
votes
2answers
50 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
2
votes
1answer
44 views

Integral inequality $\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$

Let $f(0) = 0$ and $0<f'(x)\leq1$ for all $x \geq0$, then prove: $$\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$$ The hint I was given was "differentiate, factor and ...
1
vote
0answers
29 views

Find $\lim_\limits{x\to \infty}{\sqrt[3]{x^3+ax^2+bx+c}-x}$. [duplicate]

Find $\lim_\limits{x\to \infty}{\sqrt[3]{x^3+ax^2+bx+c}-x}$. As I understand, I should use Taylor series, but I don't know how. What should I translate into Taylor series, to what extent, etc. This is ...
6
votes
1answer
62 views

Find $\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)$

Find $\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)$. I do know it is bounded by $1$. I tried using the sandwich rule with no ...
2
votes
1answer
58 views

Show that $f=0$ a.e. if $|\int_I f|^p \leq c|I|^{p-1}\int_I |f|^p$ with $0<c<1$

Suppose an extented real valued function $f$ defined on $\mathbb{R}^n$ satisfies the following two properties: a) There is a $p$, $1\leq p < \infty$ such that $f\in L^p(I)$, for every ...
1
vote
1answer
33 views

Derivatives in Topological Vector Spaces and General Spaces

I know that we may have a function $f$ such that all the directional derivative of $f$ in some topological vector space $V$, but the derivative need not exist. Moreover, it's possible for all the ...
2
votes
1answer
35 views

Sequence in an uncountable set of real numbers

Let $A$ be an uncountable subset of the real numbers, I think the following is true: There is an injective sequence $a:N\to A$ such that $\sum_{n=1}^\infty a_n$ diverges. This might also be true ...
1
vote
4answers
69 views

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$.

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$. I can't even start because I don't really know what $x\to 1^-$ means. If you know what it means it would really help me. I would as well if you ...
0
votes
1answer
52 views

Proving Submultiplicativity on a Matrix Norm

Let $||A||=(\sum_{i=1}^{n}\sum_{j=1}^{n}{a_{ij}^p})^{1/p}$, and let p=2. Then prove that $\|AB\|\le \|A\|\|B\|$ I have looked at numerous proofs for this, and I don't see one that satisfies me ...
3
votes
1answer
24 views

If $f$ and $g$ are uniformly continuous on $\Bbb{R}$ then $f\circ g$ is uniformly continuous on $\Bbb{R}$

Prove or disprove: If $f$ and $g$ are uniformly continuous on $\Bbb{R}$ then $f\circ g$ is uniformly continuous on $\Bbb{R}$. I think there's something crooked in my attempt. I would like to know what ...
1
vote
2answers
47 views

Expression for Taylor's formula with a remainder

Assume $f$ has a continuous second derivative $f~''$ in some neighborhood of $a$.Then, for every $x$ in this neighborhood, we have $f(x) = f(a) + f~'(a)(x-a) + E_1(x)$ , where $E_1(x) = \int_a^x ...
1
vote
1answer
40 views

Extending Minkowsky inequality to double summation?

I know the Minkowski inequality for sequences as follows : $$\left(\sum_{k=1}^n|x_k+y_k|^p\right)^{1/p} \leq \left(\sum_{k=1}^n|x_k|^p\right)^{1/p}+\left(\sum_{k=1}^n|y_k|^p\right)^{1/p}$$ Now say we ...
1
vote
1answer
40 views

Prove an altered p-norm is increasing

$x=(x_1, x_2, \ldots, x_n)$ Prove that $g(p)=[(1/n)(\sum_{k=1}^n |x_k|^p)]^{1/p}$ is increasing on the interval $(0, \infty)$, and find $\lim_{p\to\infty}g(p)$ I find this is extremely difficult. I ...
0
votes
0answers
10 views

Riemann Integration - Upper Sum

Let $P = \{x_o, x_1, ..., x_n\}$ and $Q = \{x_o, x_1, ...x_j, z, x_{j+1}, x_n\}$ be partitions of $[A, B]$. Note that Q is a refinement of P with just one extra point. Show that if $f: [A,B] \to ...
2
votes
0answers
46 views

Fourier transform of $e^{-\frac{x^2}{2}}\frac{1}{\mathsf{sinc}( x )} $

I have to find inverse Fourier transform of \begin{align*} F(\omega)=e^{-\frac{\omega^2}{2}}\frac{1}{\mathsf{sinc}( \omega )} \end{align*} I was wondering whether it exists maybe in a sense of ...
4
votes
2answers
120 views

Real solutions of $x^n + y^n = (x+y)^n$

I have to find all real solutions of the following equation: $x^n + y^n = (x+y)^n$ Clearly for $n = 1$, the equation holds for every $x,y$ real numbers. If $n$ is greater or equal to $2$, we do ...
2
votes
5answers
84 views

Limit problems and quandaries: finding $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$.

Find $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$. What I did is: $\lim_\limits{n\to \infty }{({n^2-n\over n^2+1})^{n+10}}=\lim_\limits{n\to \infty } {({n^2+1-1-n\over ...
1
vote
1answer
40 views

Prove that $\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$

If $f_n(x)=nxe^{-nx^2}~\forall~n=1,2,\cdots$ and $x$ real, show that $$\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$$ Attempt: By the $Mn$ Test, it ...
1
vote
2answers
31 views

Does this supremum equal infinity?

This is a generalization of the previous question Does this infinum tend to infinity? Let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuous function satisfying $$\sup ...
1
vote
1answer
31 views

Courant. Real numbers determined by nested sequences of rational intervals.

In his book Introduction to Calculus and Analysis vol.1, page 95 Courant writes: Every nested sequence of intervals with real end points contains a real number. To prove this, consider closed ...
0
votes
1answer
56 views

$ \exists c \in( a, b) \text{ such that } f(c)=\max\limits_{x \in [a, b]} f (x) $

I saw in a corrected. if We have $ f $ continuous on $ [a, b] $ with $ f (a) = f (b) $ and $ f $ differentiable left and right at $ (a,b)$ can we say that $$ \exists c \in (a,b) \text{ such that } ...
0
votes
1answer
24 views

Limsup, is there an alternative definition or am I missing the spirit of the question?

Let $X$ be the positive integers Let $H$ be $\mathcal{P}(X)$ For finite $E\in H$ $v(E)$ is the number of points in $E$. Define: ...
1
vote
1answer
34 views

nth derivative of ${1\over x}$. A problem. [on hold]

$f(x)=f^{(0)}(x)=x^{-1}$, $f^{(1)}(x)=-x^{-2}$, $f^{(2)}(x)=2x^{-3}$. Therefore, $f^{(n)}(x)=(-1)^{n}n!x^{-n-1}$. Except I see in some places that the expression is different, using, for example, ...
1
vote
2answers
40 views

IVT question involving polynomial with even degree

Let $M(x)$ be an even polynomial with a positive leading coefficient, with $a_{2n} > 0, n\ge1 $. Show that there exists a constant $a*\in \mathbb{R}$ such that $M(x)+a = 0$ has a real root if ...
0
votes
1answer
18 views

Given the supremum of a set, prove the supremum of a related set

Let S be a bounded set of positive real numbers, and let T = {s$^2$ : s $\in$ S}. Set u:= sup S. Prove that u$^2$ = sup T. Let S,T be as described. Let s$^2$ $\in$ T. Then s $\in$ S and s $\le$ u. ...
1
vote
0answers
22 views

Proof that $\sum_{n} a_{n}$ converges if $a_{n}=O(1/n)$, $\lim_{x\uparrow 1}\sum_{n}a_{n}x^{n}$ exists?

Does anyone know of a simple proof that $\sum_{n=0}^{\infty}a_{n}$ converges whenever the real sequence $\{ a_{n} \}_{n=0}^{\infty}$ satisfies these two conditions? $a_{n}=O(1/n)$; $\lim_{x\uparrow ...
0
votes
1answer
17 views

How to prove the period of a trigonometric function decreases?

Let's imagine I have a function $f(x)=\sin x$. We know that the period is going to be $2\pi$. Same idea when we think of $\cos x$. What I am saying is, that the period as $x \to \infty$ remains the ...
0
votes
1answer
19 views

What is the oscillation of a function?

Define the oscillation of a function at a point $x$ to be (for an open interval $I$): $$\omega_f(x)=\inf_{x\in I}\sup_{s,t\in I}|f(t)-f(s)|$$ I am a bit confused about the definition above. How am I ...
1
vote
1answer
25 views
+50

$U(f+g) \leq U(f)+U(g)$ proof Upper Riemann Integral

$U$ here represents the upper Riemann Integral. I understand the vast majority of this proof, however the part underlined in orange states $\forall \varepsilon>0 $ should it not be ...
0
votes
1answer
19 views

Find the extrema of a function

I'm trying to find the extrema of the function : $$ f : \bigg[\frac{1}{2},4\bigg] \rightarrow \mathbb{R}$$ $$f(x)=x^{\ln(\frac{1}{x})}$$ I tried to derivate the function in order to deduce the ...
0
votes
0answers
29 views

Does isometry preserve volume on open sets?

Suppose there are two open sets $A,B$. $h$ is an isometry. And the function $h$ maps $A$ to $B$; $h(A)=B$. I need to show that isometry is volume preserving. Any hint would be appreciated! Thanks ...
0
votes
4answers
61 views

2003 Putnam A-1 Help needed about sequences

Okay so for $n=1$ there is only one way. For $n=2$ you have, $1+1, 2 + 0$ for $n=3$ you have: $1+1+1, 1+ 2, 3 + 0$ three ways. So $P(n): n$ ways, we must prove the $P(n+1): n + 1$ statement is ...
0
votes
1answer
30 views

Using set theory to prove a function problem

I begin with: $$A = \{a \le x < x_0 | f(x) = 0 \}$$ $$B = \{x_0 < x \le b | f(x) = 0 \}$$ Let $c = \sup A$ and let $d = \sup B$ First to prove $f(x) > 0$ for $x \in (c, d)$ I will ...
1
vote
0answers
18 views

Leading up to Young's Inequality

I am trying to prove Young's Inequality by considering the function $$h(u) = \frac{u^p}{p} + \frac{C^q}{qu^q}$$ for $C,u>0$ and $p,q >1$. We also require $$\frac{1}{p}+\frac{1}{q}=1$$ so that ...
1
vote
1answer
50 views

${f \text{ is differentiable on } I \iff f_{\left|\ [a,b]\right.} \text{ is differentiable }\ \forall a,b \in I}$

Let $f\in \mathbb{R}^{I}$ $I$ interval of $\mathbb{R}$ Show that $${f \text{ is differentiable on } I \iff f_{\left|\ [a,b]\right.} \text{ is differentiable }\ \forall a,b \in I}$$ in ...
0
votes
0answers
8 views

convex region with 7 vertices

Let, $X$ be a convex region in the plane bounded by straight lines. Let, $X$ have $7$ vertices. Suppose $f(x,y)=ax+by+c$ has maximum value $M$ & minimum value $N$ on $X$ & $N<M$. Let, ...
1
vote
3answers
39 views

Find $f^{(n)}(1)$ where $f(x)={1\over x(2-x)}$.

Find $f^{(n)}(1)$ where $f(x)={1\over x(2-x)}$. What I did so far: $f(x)=(x(2-x))^{-1}$. $f'(x)=-(x(2-x))^{-2}[2-2x]$ $f''(x)=2(x(2-x))^{-3}[2-2x]^2+2(x(2-x))^{-2}$. It confuses me a lot. I know I ...
3
votes
1answer
26 views

Does there exist a non-negative valued compactly supported function such that its Fourier transform only vanishes at a given point?

My question is as follows: Given $t_0\in\mathbb{R}$. Does there exist a non-negative valued compactly supported function $f\in L^1(\mathbb{R})$ such that its Fourier transform, $\widehat f\left( t ...
1
vote
1answer
16 views

which of the following sequences $\{f_n\}\in C[0,1]$ must contain a uniformly convergent subsequence?

Could anyone tell me which of the following sequences $\{f_n\}\in C[0,1]$ must contain a uniformly convergent subsequence? $|f_n(t)|\le 3\forall t\in [0,1],\forall n$ $f_n\in C^1[0,1],|f_n(t)|\le ...
1
vote
0answers
21 views

contour integral and limit: What is the condition of the interchange the order?

In real real analysis sense, the interchange between limit and integral is hold when integrand is uniformly converges. $i.e$ \begin{align} \int \lim f = \lim \int f \end{align} Here i want to ...
1
vote
1answer
26 views

Algebra vs. Sigma-Algebra Condition

I just wanted to clarify the difference between the Algebra and $\sigma$-algebra: Algebra: If $A_1, A_2 \ldots $ are in $\mathscr A$, then $\bigcup_{i = 1}^{n} A_i \in \mathscr A$ ...
0
votes
0answers
17 views

Comparison theorem for parabolic partial differential equations

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $J\subseteq\mathbb{R}$ be an intervall $T\in(0,\infty)$ and $f\in C^0\left(\overline{\Omega}\times[0,T]\times J\right)$ be locally Lipschitz ...
1
vote
0answers
15 views

Prove there is an $a>0$ such that $\forall x\in [0,1]$, $f(x)>x+a$.

Let $f$ be continuous on $[0,1]$ and $f(x)>x\space \space \forall x \in [0,1]$. Prove there exists an $a>0$ such that $f(x)>x+a\space \space \forall x \in [0,1]$. It is really important ...
-1
votes
0answers
14 views

Prove that sup and inf of closed and bounded set is in the set.

Assume that $F$ is a non-empty, closed and bounded subset of $\mathbb{R}$, with $d(x,y) = |x-y|$. Show that $supF \in F$ and $infF \in F$.
1
vote
1answer
25 views

Transformation theorem: calculate picture of a set

I have this function: $T:(0,\infty)^2 \rightarrow T((0, \infty)^2), \quad T(x,y)=\left( \frac{y^2}{x},\frac{x^2}{y} \right)$ Now I try to estimate $T(M)$ with: $0<p<q, \quad 0<a<b$ ...
1
vote
1answer
48 views
+50

Continuity on $[a,b]$ implies uniform continuity on $[a,b]$

I don't understand the step underlined in green. I understand that for any $n$ , $|f(x_n)-f(y_n)|\geq \varepsilon$ where $x_n, y_n$ satisfy the conditions given regarding a function being not ...
1
vote
0answers
68 views

Question about the application of continuous functions and IVT

I came across a question which says: Suppose that $f:[0,2 \pi] \to \mathbb{R}$ is continuous, and $f(0)=f(2 \pi)$. 1.Show that there exists $x \in [0,\pi]$ such that $f(x)=f(x+ \pi)$. ...
3
votes
1answer
39 views

Prove that if ${\{{a_n}^2}\}$ converges (${\{a_n}\}$ is monotone), thus ${\{a_n}\}$ converges and to what?

From Fitzpatrick's Advanced Calculus book: "Suppose that the sequence ${\{a_n}\}$ is monotone, i.e., either monotonically increasing or decreasing. Prove that ${\{a_n}\}$ converges if and only if ...
3
votes
1answer
56 views

Find $\lim_\limits{n\to \infty}\{en!\}$.

Find the limit $\lim_\limits{n\to \infty}\{en!\}$. $Attempt:$ $\lim_\limits{n\to \infty}\{en!\}=\lim_\limits{n\to \infty}\{(1+{1\over 1!}+{1\over 2!}+{1\over 3!}+...+{1\over n!}+...)n!\}$. The ...