- #1
torquerotates
- 207
- 0
Limit definition gives a contradiction!
say we are given sequences a(n), b(n) such that, a(n)->a, b(n)->b
that means for epsilon>0,
a-epsilon<a(n)<a+epsilon when n>N1
b-epsilon<b(n)<b+epsilon when n>N2
set N=max(N1,N2)
when n>N,
we can subtract the two inequalities
(b-a) +0<b(n)-a(n)<(b-a)+0
the epsilons cancel
b-a<b(n)-a(n)<b-a this is a contradiction. How can a number be bigger and smaller than the same number at the same time? There must be something wrong with my logic.
say we are given sequences a(n), b(n) such that, a(n)->a, b(n)->b
that means for epsilon>0,
a-epsilon<a(n)<a+epsilon when n>N1
b-epsilon<b(n)<b+epsilon when n>N2
set N=max(N1,N2)
when n>N,
we can subtract the two inequalities
(b-a) +0<b(n)-a(n)<(b-a)+0
the epsilons cancel
b-a<b(n)-a(n)<b-a this is a contradiction. How can a number be bigger and smaller than the same number at the same time? There must be something wrong with my logic.