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 …

# Category: Mathematics Real Analysis

Mathematics Real Analysis Questions

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)!} &…

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 …

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 …

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 …

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)

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 $forall x,y in Bbb{R}$…

$g(x)$ 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.

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 $[…

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)…