a convergence sequence proof a converging (a%einc
TRANSCRIPT
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) ☐