Why why why (tiny limsup proof)

  • Thread starter Thread starter quasar987
  • Start date Start date
  • Tags Tags
    Proof
quasar987
Science Advisor
Homework Helper
Gold Member
Messages
4,796
Reaction score
32
There's a "theorem" in my book that says if x is a cluster point of {x_n}, then lim inf(x_n)\leqx\leqlim sup(x_n).

The way the author proves it is a little bit extravagant. Why not just say "Let A be the set of all cluster points. Then, lim inf(x_n)=inf(A)\leqx\leqsup(A)=lim sup(x_n)."

?
 
Physics news on Phys.org
Don't they pretty much say the same thing?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

Proving the Relationship between Lim Sup and Lim Inf
Replies
7
Views
2K
Convergence of a continuous function related to a monotonic sequence
Replies
3
Views
2K
Showing Convergent Subsequence Exists
Replies
1
Views
2K
Proof for convergent sequences, limits, and closed sets?
Replies
11
Views
2K
Inequalities Since ε is arbitrarily small, do the inequalities hold?
Replies
5
Views
2K
Problem 1-23 and 1-24 from Spivak's Calculus on Manifolds
Replies
6
Views
2K
I cant visualize/understand this convergence proof
Replies
2
Views
2K
Prove limit comparison test for Integrals
Replies
2
Views
2K
Extreme value theorem, proof question
Replies
12
Views
2K
Complex numbers sequences/C is a metric space
Replies
15
Views
2K

Hot Threads

Recent Insights

Back
Top