## \$lim_{xrightarrowinfty} e^{-x^2}e^{2x}\$ (epsilon-delta proof)?

I am struggling to develop an epsilon-delta proof for the following: \$lim_{xrightarrowinfty} e^{-x^2} e^{2x} = L\$ (unknown, believed to be 0) I am aware that to do so, we must show \$existsbeta …

## Closure of \$ell^2\$ in the space of real sequences

Using the product topology on \$overline{mathbb{R}}^omega\$, is \$ell^2\$ (the space of real square summable sequences) a dense subset of \$overline{mathbb{R}}^omega\$ ?

## Let S be as set of real numbers , and let \${x_n}\$ be a sequence which converges to l

Let S be as set of real numbers , and let \${x_n}\$ be a sequence which converges to l. Suppose that for every \$n inmathbb{N},x_n\$ is an upper bound for S . prove l is an upper bound of S And also …

## Show that certain lim sup additivity implies convergence.

I’m trying to prove the following: Let \${a_n}\$ be a bounded sequence. If for every bounded sequence \${b_n}\$ the following holds: \$\$1) limsup_{n to infty} (a_n + b_n) = limsup_{n to infty} …

## Continuity of \$argmax\$ of a strictly concave function

how would you show that \$\$f(x) = argmax_{yinmathbb{R}}{ay+bx+c-left|left|y-xright|right|^2}\$\$ is continuous? It is well defined since the expression under argmax is strictly concave and thus …

## How to sketch image under function on given set

For the function, \$f(r,theta)= (rcostheta , rsintheta )\$, I want to sketch the image under \$f\$ of the set \$S=[1,2]\$ x\$ [0,pi]\$ My first step was to find the images of \$f\$ along the borders of …

## The boundary of a domain without a closed subset

Let \$D\$ be a bounded domain in \$mathbb{R}^{N}\$ (\$Ngeq2\$) and \$E\$ a closed subset of \$D\$ with empty interior. Show that the boundary of \$Dsetminus E\$ is the union of \$E\$ and the boundary of \$D\$: \$\$…

## Schwarz inequality for multiple integrals: \$left[int_A f(x)g(x) dxright]^2 le int_A f(x)^2 dx cdot int_A g(x)^2 dx\$

If \$f,g:Ato mathbb{R}\$ are integrable, prove the Schwarz inequality \$\$left[int_A f(x)g(x) dxright]^2 le int_A f(x)^2 dx cdot int_A g(x)^2 dx\$\$ This is that type of question that …

## Substitution rule in complex analysis

Let \$lambda_{mathbb{C}}\$ be the Lebesgue measure in the complex plane. Let \$f\$ be an entire function and \$g\$ a continuous function. I ask: When does the substitution rule hold \$\$int_{mathbb{C}}…

## Prove that \$underline{int_A}{f(x) dx} + underline{int_A}{g(x) dx} le underline{int_A}{[f(x) + g(x)] dx}\$

Question: Let \$f,g:Atomathbb{R}\$ bounded in the set \$A\$. Prove that a) \$\$underline{int_A}{f(x) dx} + underline{int_A}{g(x) dx} le underline{int_A}{[f(x) + g(x)] dx}\le …