R.A. Arts, Shri M.K. Commerce and Sri S.R. Rathi Science College, Washim

Submitted ByKu. Nikita Santosh

Bhalerao

Year 2015-2016

B.Sc :- II ( Third Semister )

Presented To

Jaju Madam And Gattani Madam

SEMINAR ON

SEMINAR ON

MATHS

Theorem :-A seq <Sn> Converges if & only if it is a Cauchy Sequence.

Sn

Convergent s

nL

lim

L .Exist is Sn Seq of Lim

DefinationBy

iSn Seq -:Part IF

Couchy Criterian for Convergents

By Defination

For Є > 0 Э integer M Such That

Sm

& Sn

mn

n

n

L

L

lim

sConvergent is Sn Seq

lim

(1)2/ L -Sn

By Defination

For Є > 0 Э integer M Such That n ≥ M

)2(2 L - Sm /

For Є > 0 Э integer M Such That m,n ≥ M

Couchy Converges isSn Seq

Sn - Sm

(3) & (2), (1), From

)3(Sn - L L - Sm

Sn -L L - Sm Sn - Sm

, Naw

Only If Part :-

If Seq < Sn > is a Cauchy Sequence By Defination

For Є > 0 Э integer M Such that

m, n ≥ M

M = n+1n > m

Sn - Sm

) DecresingOr IncresingEither (

Monotonic is Sn Seq

) DecresingOr IncresingEither (

Monotonic is Sn Seq

But, We Know

Every Cauchy Sequence is always Bounded Seq < Sn > is Monotonic & bdd

&

By Theorem

Seq is < Sn > is Convergents Hence

Thank you……..!!!