Limit of the Sum of Two Sequences
Limit of the Sum of Two Sequences
This Demonstration illustrate the proposition (+)=+. Let =+, , , , and . Choose positive integers and so that for all and all , and . Then for all , . Therefore, =c=a+b.
lim
n∞
a
n
b
n
lim
n∞
a
n
lim
n∞
b
n
c
n
a
n
b
n
a=
lim
n∞
a
n
b=
lim
n∞
b
n
c=a+b
ϵ>0
M
L
m≥M
l≥L
|a-|<ϵ/2
a
m
|b-|<ϵ/2
b
l
k≥K=max[m,l]
|c-|=|a+b-(+)|=|a-+b-|≤|a-|+|b-|<ϵ/2+ϵ/2=ϵ
c
k
a
k
b
k
a
k
b
k
a
k
b
k
lim
n∞
c
n