Can Polynomials Approximate a Bounded Function on the Entire Real Line?
According to the Stone-Weierstrass theorem, a polynomial can approximate a continuous function on a closed interval.I agree that, a polynomial cannot approximate a function that is unbounded on an open...
View ArticleMonotonically increasing function and supremes
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a monotonically increasing function and $A \subset \mathbb{R}$ where $A \neq \emptyset$ and boundend.i) If f is continuous function, prove that $f(\sup (A))=...
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Hardy´s "A course of pure mathematics" 10th edition, problem 31 miscellaneous...
This question has already been asked before, but has failed to generate any sufficient response. A comment on the other post suggests the problem might be incorrect. I'd really appreciate some...
View ArticleSummable functions and countable sets [closed]
If $f:X \to \mathbb{R}$ is summable function, then the set $$A=\{x \in X : f(x) \neq 0\}$$ must be countable.
View ArticleHow to solve $e^x = -x$ [duplicate]
Solve $e^x = -x$.A friend told me that there are no algebraic solutions. Is that the case? If so, how can it be shown that there is no algebraic solution?
View Articlelog inequality about monotonicity [closed]
Is it true that$\int_{k-1}^{k} \log (x)dx < \log (k)< \int_{k}^{k+1} \log (x)dx$ can you prove me that? I need this to prove Stirling.
View ArticleExpectation of Maximum and Minimum of Asymptotic Exponentials
Suppose we have $K$ series of positive random variables $\{X_{k,n},k=1,\dots, K, n\geq 1\}$. Series are independent of each other, but for each $n$, $X_{k,n},k=1,\dots, K$ are identically...
View ArticleIf two sequences $\{a_n\}$ and $\{b_n\}$ be such that $a_n>b_n$ then show...
$$ \text{Take two converging sequences } \{a_n\}_{n=1}^\infty \text{ and } \{b_n\}_{n=1}^\infty, \text{ such that } a_n > b_n \text{ for all } n. \text{ Show that } \lim_{n \to \infty} a_n \geq...
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Help with this question about rank of Hessians
I'm trying to solve this question below:I need a hint to begin to solve this question.
View ArticleFind Lower bound given differential inequality [closed]
I was reading some texts online and i found this one intriguing:Let $u$ be a real $C^1$ function defined on $[0,\infty)$ such that $ \partial_x u <u-\frac{2}{u}-1$ for every $t$. Find a $C>0$...
View ArticleIf $\lim_{n \to \infty}x_n=x$ , then $\lim_{n \to \infty} \left(\lfloor...
If $\lim_{n \to \infty}x_n=x,$ where $x$ is any real number, then $\lim_{n \to \infty}\left(\lfloor x_n^2\rfloor+\lfloor x_n\rfloor\right) =?,$ where $\lfloor y\rfloor$ denotes the greatest integer...
View ArticleWhy can't I solve $\int y' y'' dx$ changing the order of diff and integration?
I know that in order to solve: $\int y' y'' dx$I can do $dy' = y''dx$, which is the same as subing $u=\frac{dy}{dx}$ and $du=\frac{d^2y}{dx^2} dx$ which results in:$\int y' dy' = \frac {y'^2}{2} +...
View ArticleConsidering a limit on subsequences of reals, instead of all reals
Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a function satisfying$$\lim _{x \rightarrow \infty} \frac{f(x)}{x^k}=a \in \overline{\mathbb{R}}, \quad k \in \mathbb{N}^* \backslash\{1\}$$and$$\lim _{x...
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What's wrong with this "proof" that $2=\sqrt{2}$? [duplicate]
Suppose you were walking from the top left corner of a large square $S$ with side length $1$ to the bottom right. To do so, you go to from the top left corner to the center of square $S$ in a right...
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Riemann Integral and Rational Numbers
Hello,I am self-studying real analysis and I am unable to solve this very interesting question I came across.I am not fully sure why the upper Darboux sum for this function is equal to 0. The argument...
View ArticleInjectivity of $L2$ Operator with Delta Distribution in Frequency Domain
I'm trying to understand if an operator $T: L^2(\mathbb{R}) \to H$ is injective. The operator $T$ is well-defined in the sense that $H$ is defined as the image of $L^2(\mathbb{R})$ under $T$. We have a...
View ArticleLebsesgue integrable functions as the closure of Riemann integrable functions
I've read that Lebesgue integrable functions can be thought of as the "completion" of Riemann integrable functions. I'm curious about the specific norm or metric in which this closure is...
View ArticleA sequence problem on the sum of powers of two sequences [closed]
I'm having trouble solving this sequence problem: let $p,q $ be odd integers and $u_n$, $v_n$ be real sequences, such that $u_n+v_n \to 0$ and $u_n^p+v_n^q \rightarrow 0$ as $n \to \infty$.Show that...
View ArticleUnderstanding notation of a function space from Nestorov's Convex...
I am looking at Theorem 2.1.5 in the book Lectures on Convex Optimization, by Yuri Nestorov, where the theorem statement is for all $f\in \mathcal{F}_L^{1,1}(\mathbb{R}^n, \|\cdot\|)$ and I don't...
View ArticleAn inequality for norm in Hilbert space
I read the following in a paper:Let $H$ be a real Hilbert space. For $y, k \in H$, and $p \geq 2$, one has$$\bigg| |y+k|_H^p - |y|_H^p - p |y|_H^{p-2} \langle y, k\rangle_H \bigg| \leq...
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