Sequential criterion for limits and axiom of choice
This may be an absurd question but it doesn't let me sleep. Why doesn't the usual proof of the sequential criterion of (real) limits requiere the axiom of (countable, in that case) choice? A professor...
View ArticleTheorems for interchanging limit and conditionally convergent integral
I'm trying to prove that $$\lim_{s \to \infty} \int_{0}^{\infty}\frac{\sin(x)}{x}x^{\frac{1}{s}} dx = \int_{0}^{\infty}\frac{\sin(x)}{x} dx = \frac{\pi}{2}$$ by exchanging the limit and the...
View ArticleQuestion about the exercise 2.5.50 of Folland's textbook
Suppose $(X, \mathcal{M}, \mu)$ is a $\sigma$-finite measure space and $f \in L^{+}(X)$ where $f: X\to [0,\infty]$ are all measurable functions. Let $$G_f = \{ (x,y) \in X \times [0, \infty]: y \le...
View ArticleGradients of two functions with the same level sets are parallel for all...
Is the intuition behind this statement, based on : 1. The definition that a gradient is perpendicular to the level curves2. Since the level sets are the same for both functions, the corresponding...
View ArticleFinding maximum of function of $n$ variables.
Let $a,b \in \mathbb{R}$ such that $0<a<b$and let $n \in \mathbb{N}$. Consider the function$$f(x_1,...x_n)=\frac{(x_1\cdot... \cdot x_n)}{(a+x_1)(x_1+x_2)...(x_{n-1}+x_n)(x_n+b)}$$ where $x_i \in...
View ArticleBolzano-Weierstrass Theorem is false when $S\subseteq\mathbb{Q}$
Show that the Bolzano–Weierstrass theorem is false when $S \subseteq \mathbb{Q}$. The Bolzano-Weierstrass theorem states that if a set $S\subseteq\mathbb{R}$ is infinite and bounded, it has an...
View ArticleUniform Convergence of Integrable Functions and Counterexamples [closed]
Let $ \{ f_n \}_{n=1}^{\infty} $ be a sequence of real-valued integrable functions on $ X $ such that$\sup_{x \in X} |f_n(x) - f(x)| \to 0 \quad \text{as } n \to \infty$(that is, $ f_n $ converges to $...
View ArticleCan at least one fixed point of a correspondence be approximated by fixed...
Let $X$ be a non-empty compact convex subset of $\mathbb R^n$, where $n$ is a positive integer. Let $\Gamma$ be a correspondence that maps from $X$ into $X$—meaning that for every $𝑥\in X$, $\Gamma(x)$...
View ArticleHow to show a function is continuous [closed]
So I am given the function $f(x)=x^{69420}$ on $\Bbb{R}$ where $\Bbb{R}$ is the real Numbers. How do I show that $f(x)$ is continuous? (I am stuck on this).My attempt: So I know the definition of a...
View ArticleProof of Orthonormal basis for $L_2(\mathbb R)$ by Hermite Polynomials
Consider $L^2(\mathbb{R})$ as an Hilbert space with inner product $(\cdot ,\cdot)$. Define$$\psi_n(x)=e^{-\frac{x^2}{2}}H_n(x),$$where $H_n(x)$ is the Hermite Polynomials. How do you show that...
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Why the auxiliary function in Taylor's theorem proof is this?
I want to know the reasoning why the auxiliary function is in such form.A couple examples:On the OpenStax website there is a Calculus Volume 2 book. The proof of Taylor's theorem with remainder in...
View ArticleSum of a complex function being real
Suppose $f:\mathbb{C}\to\mathbb{C}$ is a complex function, and that $a_1,\cdots,a_k\in\mathbb{C}$ are complex roots of another function (it doesn't matter what it is for the purposes of this question)....
View ArticleWhat's the big idea / motivation between making a non-autonomous ODE autonomous?
There are a few posts on this forum that deals with the transformation of non-autonomous differential equations into autonomous differential equations, in particular: Transforming differential...
View ArticleHow to integrate $\int \frac{\cos x}{\sqrt{\sin2x}} \,dx$?
How to integrate $$\int \frac{\cos x}{\sqrt{\sin2x}} \,dx$$ ?I have:$$\int \frac{\cos x}{\sqrt{\sin2x}} \,dx = \int \frac{\cos x}{\sqrt{2\sin x\cos x}} \,dx = \frac{1}{\sqrt2}\int \frac{\cos...
View Article$X_n \to X, Y_n \to c$ in distribution implies $X_n Y_n \to Xc$ in distribution
I am trying to prove $$X_n \xrightarrow{d} X, Y_n \xrightarrow{d} a \implies Y_n X_n \xrightarrow{d} aX$$where $a$ is a constant.What I tried:Let $g:\mathbb R\to \mathbb R$ an arbitrary uniformly...
View ArticleEvaluate $\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$
Evaluate the following integral$$\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$$This function does not have an elementary anti-derivative. How can we solve this?
View ArticleExercise 3.2.11 of Garling (A Course in Mathematical Analysis, Volume 1)
I'm currently working on part (a) of exercise 3.2.11 in Garling's book, and I'm posting a question because I'm having trouble.Problem$n\in\mathbb{N}$, $t\in\mathbb{R}$, if $nt > -1$, then $(1-t)^{n}...
View ArticleExpressing the zeta function as a sum of primes
We begin with the well-known Euler product form of the Riemann zeta function, an alternative to the summation definition. We can then expand it into a series:$$ \zeta(s) = \sum_{n=1}^{\infty}...
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Does Uncertainty Principles apply to solutions of finite duration (so, to...
Does Uncertainty Principles apply to solutions of finite duration (so, to something it stops moving in finite time)?I have incorporated the calculations I would like to reproduce for Gabor's...
View ArticleCan we evaluate the generalized Ahmed integral...
Does anyone have any idea on how to evaluate the following generalized Ahmed integral?$$I(t):=\int_{0}^{1}\frac{\arctan(t\sqrt{2+x^2})}{\sqrt{2+x^2}(1+x^2)}\,dx$$Here is my...
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