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Mathematics Real Analysis Questions
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 …
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. 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 …
I’m trying to prove the following: Let ${a_n}$ be a bounded sequence. If for every bounded sequence ${b_n}$ the following holds: $$1) limsup_{n to infty} (a_n + b_n) = limsup_{n to infty} …
how would you show that $$f(x) = argmax_{yinmathbb{R}}{ay+bx+c-left|left|y-xright|right|^2}$$ is continuous? It is well defined since the expression under argmax is strictly concave and thus …
For the function, $f(r,theta)= (rcostheta , rsintheta )$, I want to sketch the image under $f$ of the set $S=[1,2]$ x$ [0,pi]$ My first step was to find the images of $f$ along the borders of …
Let $D$ be a bounded domain in $mathbb{R}^{N}$ ($Ngeq2$) and $E$ a closed subset of $D$ with empty interior. Show that the boundary of $Dsetminus E$ is the union of $E$ and the boundary of $D$: $$…
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 …
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 $$int_{mathbb{C}}…
Question: Let $f,g:Atomathbb{R}$ bounded in the set $A$. Prove that a) $$underline{int_A}{f(x) dx} + underline{int_A}{g(x) dx} le underline{int_A}{[f(x) + g(x)] dx}\le …