let {xn} be a sequence that does not converge and let N be any real number> prove that there
exist >0 and a subsequence {xpn} of {xn} such that the absolute value of xpn - L > for all n
Solution
A sequence (xn) in X converges to x if for every neighbor-hood U of x there exists N N such
thatxn U for all n N.
In this case we writexn x or limn!1xn = x or limnxn = x.A
sequence (xn) which does not converge is called divergent.
The following are equivalent.(i) limn xn = x.(ii)
For every \" > 0, there is N N such that xn B(x, \") for all n N.(iii) For every \" > 0, there is N
N such that d(xn, x) < \" for all n NProposition
(i) (Uniqueness of the limit) A sequence cannot havemore than one limit.(ii) (Boundedness) If
(xn) converges to x, then.
let {xn} be a sequence that does not converge and let N be any real .pdf
1. let {xn} be a sequence that does not converge and let N be any real number> prove that there
exist >0 and a subsequence {xpn} of {xn} such that the absolute value of xpn - L > for all n
Solution
A sequence (xn) in X converges to x if for every neighbor-hood U of x there exists N N such
thatxn U for all n N.
In this case we writexn x or limn!1xn = x or limnxn = x.A
sequence (xn) which does not converge is called divergent.
The following are equivalent.(i) limn xn = x.(ii)
For every " > 0, there is N N such that xn B(x, ") for all n N.(iii) For every " > 0, there is N
N such that d(xn, x) < " for all n NProposition
(i) (Uniqueness of the limit) A sequence cannot havemore than one limit.(ii) (Boundedness) If
(xn) converges to x, then
