Sirawich Saranakomkoop 062101840

A convergence sequence in Cauchy but the converse is not always true.Proof

consider a converging sequence (a%eiNc ✗⇒ 7- a c- ✗ such that V-E>0

,7- PEN

with IAN- of / ⇐ E- for any NZP .Form > n , then one has

IAN- am I =/ aka•+of- am /

⇐ I aka•

I + 1am- amC- Ez + G- = E for N, M > P

Therefore, convergence sequence is in Cauchy sequence .

Show that Cauchy sequence is not always a convergence sequence .Given co# { Can)next ¥1. an --0}and ECE)={ Can)n⇐⇐1 an -1-0 for a finite number of n }

From definition,Cote c co (E)

Let ( ai ) nee E CORD and fix N

air = { In / +1if Ink N. i.

o if 1m17 N-

•

µ...

••

It•

••

-

. .

• µ÷!¥÷•n-it :For any Can)nez

Itani 11 -_ max tantNEZ

Let N >M ⇒ Hakam 1k¥,

Prove that Layneµ is Cauchy in oct) but not converging in Cote

V-E>0,fix P c- IN with P > Iq

For N >ME P

one has

11 a"- am 11 =# < In ⇐ f- < e⇒ ( a%eµ is Cauchy in Cc (a)Let us set

AT = In /+ ,ttn EZ

Then

1.) (AP)nez E co (2)¢ cc (2)

since an to for a finite number of n

2) The sequence (a%c⇒ convergesto a" in co (2)

,not in Cda)

Indeed ,for any E> o

choose PEN with P>&Then for N > P

one has

" a"a%= # < * <* < ,•÷

Co (2)

⇒ ( AN / new converges to of ( as is in coth,not in Coto)

Thus, Cauchy sequence in co (2) does not always converge

in co (2), but converges only in co (2) ☐