## Let \$S subseteq mathbb{R}\$ have a supremum \$x\$. Show that there is a sequence \$(x_n)\$ in \$S\$ that converges to \$x\$.

I understand how to prove this using the approximation property of the supremum and invoking the squeeze theorem of limits. However, upon reading answers for similar questions, I think there’s a more …

## \$P_3(x)\$ and \$R_3(x)\$ of \$f(x)=e^{-3x}+3 sin (x)-1\$

the maclarin series for \$sin(x)\$ is \$\$ sin(x)=sum^{infty}_{k=0} frac{(-1)^k*x^{2k+1}}{(2k+1)!}\$\$ so summation to \$k=3\$ \$\$ begin{aligned} sum^{k=3}_{k=0} frac{(-1)^k*x^{2k+1}}{(2k+1)!} &…

## Estimating the error for \$e-(1+frac 1n)^n\$

I would like to estimate the error \$e-(1+frac 1n)^n\$, for arbitrary \$n ge 0\$. I think I have found a way to do it, but it does not really seem optimal to me. Here is what I did: I considered the …

## Is this implication correct?

Let \$a\$ and \$b\$ two coprime integer numbers. Then there exist \$u₁\$ and \$v₁\$ such that \$\$au_1+bv_1=1tag1\$\$ Let us consider an equation of the form: \$\$ah+bd=gtag2\$\$ where \$h\$ is an unknown …

## Munkres’ Proof of Well Ordering Property

In Munkres’ Topology the proof for the Well-Ordering Property is stated as follows: I’m having confusion with the first and second underlined part: “Let A be the set of all positive integers n for …

## How to show \$inf B\$ is fixed point?

This question is from here: I want to show that \$z=inf B\$ is one of the fixed points of \$h\$. I have done this: Proof(Partial): If \$zin B\$, then \$h(z)le z\$. If \$h(z)

## Uniform contiunuity of \$f\$?

If \$g \$ is uniformly continuous and \$g(x) = (f(x))^2\$,\$f(x) geq 0\$, then is \$f\$ uniformly continuous? So, \$forall epsilon > 0 , \$ there exists \$delta > 0\$ such that \$forall x,y in Bbb{R}\$…

## How to show \$g(x)=frac{4^x+x^2-ln(2)cdot x-1}{tan(2x)}\$ is continuous at \$x = 0\$ for \$g(0) := ln(sqrt{2})\$

\$g(x)\$ is defined on the Intervalls \$(frac{-pi}{2}, 0) cup(0,frac{pi}{2})\$ I’ve tried doing it by using L’Hôpital but \$sin(x)\$ gets into the denominator and I can’t get rid off it.

## Question about a proof of the Cauchy Criterion for Riemann Integrability

The Cauchy Criterion is typically stated as follows: \$f\$ is Riemann-Integrable on \$[a,b]\$ if and only if for every \$epsilon>0\$, there is a \$delta>0\$ so that for any two tagged partitions of \$[…

## What does \$Bbb S^{n-1}times mathbb{R}\$ stand for?

Let \$ngeqslant 1,f: Bbb R^n – { 0 } to S^{n – 1} times Bbb R,xto(frac{x}{||x||},ln(||x||))\$ is a homeomorphism which the inverse is \$f^{-1}:S^{n-1}timesmathbb{R}tomathbb{R}^n-{0},(y,t)…