Recent content by zigzagdoom

Thank you for the help

Thanks Samy, I did not realize a sequence can converge to a Limit from both the left and the right - will keep that handy fact in mind. So in terms of wording I would approach as follows; ##∀\epsilon > 0, ∃N ∋## ##∀n,m > N, | s_n - L | < \epsilon## and ##| s_m - L | < \epsilon##. By the...

OK so from the above, I would do as follows: Third Attempt Since ##|s_n-L| < \epsilon## ##|s_n| < |L + \epsilon|## ##|s_n - s_m| < |L + \epsilon - s_m|## ##| (s_n - L) - (s_m - L) | < |L + \epsilon - s_m|## But ##|s_n-L| < \epsilon## and ##|s_m-L| < \epsilon## as the both ##s_m## and...

Thanks, Would the argument I posted before be incorrect? I essentially just used the supremum as a point of reference to gauge distance between points.

Attempt A sequence is called a Cauchy sequence if the terms of the sequence eventually all become arbitrarily close to one another That is, given ε > 0 there exists N such that if m, n > N then | am - an | < ε (sn) is convergent to L if for every ε > 0, ∃N such that ∀n > N, | sn - L | < ε...

The Attempt at a Solution Suppose (sn) is a convergent sequence with limit L. Let S = {sn: n∈ℕ} By Theorem 1, every convergent set is bounded. By Theorem 2, sup S exists, let Sup S = M. Now, since M = Sup S, consider some M - φ < M for some 0 < φ < ε. But then this means ∃sn such that M - φ...

Thanks LCKurtz, I think i implcitly assumed that the sequence is monotonically increasing. But the Theorem 3 is not true as a a sequence denoted by the sine function would be bounded, yet not convergent I suppose. I will give it another crack

Hi, I am trying to prove that every convergent sequence is Cauchy - just wanted to see if my reasoning is valid and that the proof is correct. Thanks! 1. Homework Statement Prove that every convergent sequence is Cauchy Homework Equations / Theorems[/B] Theorem 1: Every convergent set is...

Would anyone know of a hard computational multivariable calculus book (i.e. not a real analysis type proof based book).? Stuff with hard integration questions or deeper algebraic manipulations would be especially useful. Primary aim is to use the text to study mathematical methods. Thanks!

Thanks a lot for this - guidance has been very useful in helping me think about the problem.

Second attempt at proof Suppose that x is an accumulation point of a set S. Then for every ε > 0, it is the case that N * ( x; ε ) ⋀ S ≠ Ø. Now consider any y such that y ∈ N * ( x; ε ) ⋀ S. Since N * ( x; ε ) ⋀ S ⊆ S \ {x} ,we know that S \ {x} ≠ Ø. Now let a sequence of a number of y be...

. Thanks again. The precise definition that I have in the book is: "x is an accumulation point of S if every deleted neighbourhood of x contains a point of S. That is, for every ε > 0, N * ( x;ε ) ≠ ∅." N * ( x;ε ) ≠ ∅ is the deleted neighbourhood of x.

Thanks, In response to your questions: x is an accumulation point if N(x;ε) ∧ S ≠ ∅. S is not defined, but I have taken it to mean some metric space (the metric has not been specified). The space of the question has not been specified. I have assumed x is an accumulation point, so that N(x;ε) ∧...

Thanks I will rework and see if I can incorporate this

Hi guys, I attempted to prove this theorem, but just wanted to see if it a valid proof. Thanks! 1. Homework Statement Prove that x is an accumulation point of a set S iff there exists a sequence ( s n ) of points in S \ {x} that converges to x Homework Equations N * ( x; ε ) is the x -...