Question

what is cauchy's first theorem on limits

Answer 1

Given the length of the space continuously, at every instant of time], to find the speed of motion the derivative] at any time proposed.

Answer 2

Lol this has to be Calc, good luck!

Answer 3

it just says that if the sequence converges then at some point every point after that point the values are arbitrarily close to each other

Answer 4

@zzr0ck3r Way better way to put it than I did.

Answer 5

I have no idea what you said.

Answer 6

cauchy's first theorm on limits says that if lim n tends to infinity and s(n) tends to 'l' then as n tends to infinity s(1)plus s(2) upto so on upon 'n' tends to 'l'

Answer 7

right

Answer 8

at some point its always less than some "epsilon"

Answer 9

do you have its proof

Answer 10

im sure its in one of my book's.

Answer 11

sec

Answer 12

we must be thinking of two different theorems. Cauchy sequence is a definition. And Cauchy theorem states that a sequence is Cauchy iff it converges

Answer 13

this one i can prove

Answer 14

are you talking about convergent sequences in class?

Answer 15

can you send me its proof somehow

Answer 16

its not that bad if I remember right, do you know bolzano-weirstrass theorem?

Answer 17

yes the theorem includes convergent sequences

Answer 18

no I dont know

Answer 19

hey man are you going to help me or not

Answer 20

suppose {x_n} is cauchy. given a=1, choose N natural such that |x_N-x_m|<1 for all m>=N.
then
|x_m|<1+|x_N| for m>N
so {x_n} is bounded by max{|x_1|,|x_2|,.....}
by the BW theorem x_n has a convergent sub sequence
we can call it {x_n_k} that goes to c for large k
let v>0
x_n is cauchy so choose N_1 natural s.t.
n,m>N_1 imples |x_n-x_m|<b/2
since x_n_k goes to c for large k
choose N_2 s.t. k>=N_2 imples |x_n_k-c|<b/2
fix k>N_2 s.t. n_k >N_1
then |x_n-c|<=|x_n-x_n_k|+|x_n_k-c|<b
for all n>= N_1
so x_n goes to c as n gets large

Answer 21

let me know..the first bit is all about proving its bounded so we can aply BW thm

Answer 22

apply

Answer 23

"hey man are you going to help me or not"

could you not tell that I was typing?

Answer 24

its really just a different definition convergent sequence, and they both say the same thing.

Answer 25

I havnt read about cauchy's sequences the only thing about which I read is convergent and bounded sequences and now I am on this theorem so if you could help me? or else atleast tell me what are cauchy's sequences

Answer 26

A sequence {x_n} is said to be Cauchy (in R) iff for all e>0 there exists N natural s.t. n,m >=N implies |x_n-x_m|<e

Answer 27

okay thanks bye

Answer 28

bye