Failing to understand why the limit is tending to $+\infty$
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be such that there exists $M>0$ satisfying$$\left|f\left(x_1+\cdots+x_n\right)-f\left(x_1\right)-\cdots-f\left(x_n\right)\right| \leq M, \quad \forall x_1,...
View ArticleProving that the second derivative of a convex function is nonnegative
My task is as follows:Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function, and let $f$'s second derivative be continuous. Let $f$ be convex with the following definition of convexity:...
View ArticleDoubt in Convergence in Distribution . [closed]
The Limit funtion in the definition of Covergence in Distribution$$lim_{n\to\infty} F_{n}(x)=F(x)$$It is said "F(x) or the limit funtion may or may not be Distibution function".I need the reason behind...
View ArticleIf $\{x_n\}$ converges to a non zero positive number then show that...
Although a proof of this claim is provided here my question is whether I can prove this result this as follows?My proof$$\text{A sequence, $\{a_n\}$, is said to diverge to $\infty$ if for any given...
View Articlewriting the real line as a countable union of measure zero sets. [closed]
I am thinking about writing the real line as a countable union of measure zero sets.Is it just writing it as a countable union of singletons? i.e., $$\mathbb R = \cup_{x\in \mathbb R}\{x\}$$Is this the...
View ArticleHow to prove that nth root of natural number exists in real numbers.
Nth root of a number A is a real number that gives A, when we raise it to integer power N.This is how Nth root is defined in the book I am learning fromI have seen the proof of how √2 and cube root of...
View ArticleExistence for a simple proof: $\ln(x)$ is concave on its domain
I was reviewing a bit of analysis, and for fun I thought to prove $f(x) = \ln(x)$ is concave on its domain just by using the definition of concave function. After having tried a bit, getting stuck in a...
View ArticleAsk an identity on inner products
Question: how to get the the identity$$\left\|y_n-y_m\right\|^2=2\left\|x-y_n\right\|^2+2\left\|x-y_m\right\|^2-4\left\|x-\frac{y_n+y_m}{2}\right\|^2$$The background of the question is from the book...
View ArticleShow that the sequence $a_n = \frac{3n^4 - 5n^3 - 9n^2 + n -3}{n^4 -3n^3}$...
Its is obvious that the given sequence converges to 3 but the problem is how do i show this using the epsilon delta definition. I have two approaches but i get stuck in both of them midway.Approach...
View ArticleNeed examples of problems of the form "Construct a function $f:[0,1]...
Let $I$ be the unit interval. I developed a method to simplify problems of the form "Construct a function $f:I \rightarrow I$ satisfying given properties". These properties should only be expressible...
View ArticleAn Interesting Integral of Weight 4 (from Pisco)
An interesting integral is given in this post by Pisco, that is$$\int_0^1 \frac{x}{1+x^2}(\log x)(\log(1+x))^2 dx $$$$= -\frac{15 \text{Li}_4\left(\frac{1}{2}\right)}{8}-\frac{7}{4} \zeta (3) \log...
View ArticleExercise on Borel-measurability
Let $A$, $B \in \mathcal{B}(\mathbb{R})$, $y_0 \in B$ be an accumulation point for $B$, and let $f: A \times B \to [-\infty, +\infty]$ be a $\mathcal{B}(A \times B)$-measurable function. Assume that...
View ArticleProve that $\ell^{p}(\mathbb{N})$ is a Banach Space [duplicate]
Let $l^p(\mathbb{N})=\left\{ \{x_n\}_{n=1}^{\infty} : \|x\|_p=\left(\sum\limits_{n=1}^{\infty}|x_n|^p\right)^{1/p} < \infty \right\}$ with $1 \leq p < \infty$.I would like some insight on how to...
View ArticleDoes the Space of Functions have a Compact Covering?
Let $X$ be an arbitrary set and $\mathbb{R}^X$ denote the set of real valued functions on $X$. Define the metric $$d(f,g) = \sum_{i=1}^\infty \frac{i \wedge \sup_{x \in X} |f(x) - g(x)|}{2^i}$$ on...
View ArticleQuestions on derivative of absolutely continuous function
Let us consider an absolutely continuous function $f:[0,h] \to \mathbb{R}$.Then, it is well-known that $f'(t)$ exists almost everywhere for $t \in [0,h]$.What I wonder to check if it is true is the...
View ArticleSpivak Problem 3-23
The problem reads, with $A \subset \mathbb{R}^n$ and $B \subset \mathbb{R}^m$ for some $n, m \in \mathbb{Z}^+$:Let $C \subset A \times B$ be a set of content $0$. Let $A' \subset A$ be the set of all...
View Articlehow to prove that $\sin x+\sin \pi x$ do not have maximum and minimum values
That's my line of thinking，I feel like there's still something wrong with itGiven that $f(x) \leqslant 2$, if there exists $x_0$ such that $f\left(x_0\right)=2$, then it must be true that $\sin x_0=1$...
View ArticleDoubts over the Second fundamental theorem of calculus
I've read in many places two different definitions of the second FTC, I'm interested about the general form:As presented hereIf $F$ is differentiable on $[a,b]$ and the derivative $F'=f$ (say) is...
View ArticleBound an integral from below
the following question appeared in my calc exam:prove that there exists some $c>0$ such that for all $n \in \Bbb N$,$$cn \le\int_0^n f(x)dx$$whereas $f(x) = x^2 - \lfloor x^2 \rfloor$I've tried...
View ArticleProof of Fubini's theorem for infinite sums.
In his book Analysis 1, the author Tao states Fubini's theorem as follows Let $f:N \times N \rightarrow \mathbb{R}$ be a function such that $\sum_{(n,m)\in N\times N}f(n,m) $ is absolutely convergent....
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