Prove or disprove that if (x_n) is convergent and lim(x_n)<2 then ∃ M∈N and c>0 such that when n≥M, x_n<2-c
Any help would be greatly appreciated! Thanks =D
Any help would be greatly appreciated! Thanks =D
-
Assume that the limit of {a[n]} equals L < 2. Let c = (2-L)/2 >0. Then, by the definition of L, there exists M such that for any n>M, |a[n] - L| < c, or equivalently, -c < a[n] - L < c.
So, consider a[n] - L < e:
a[n] < L + c = 1 + L/2 = 2 - (2-L)/2 = 2 - c.
.
So, consider a[n] - L < e:
a[n] < L + c = 1 + L/2 = 2 - (2-L)/2 = 2 - c.
.