Linear Algebra, Theory And Applications, 2012a

440 COMPACTNESS AND COMPLETENESS
D.0.2
Convergent Sequences, Sequential Compactness
A mapping f : {k, k +1,k+2, ···}→R p is called a sequence. We usually write it in the
form {a j } where it is understood that a j ≡ f (j).
Definition D.0.7 Asequence,{a k } is said to converge to a if for every ε>0 there exists
n ε such that if n>n ε ,then|a − a n | n ε,then
|a − b| < |a − a n | + |a n − b| < ε 2 + ε 2 = ε.
Since ε is arbitrary, this proves the theorem.
The following is the definition of a Cauchy sequence in R p .
Definition D.0.10 {a n } is a Cauchy sequence if for all ε>0, thereexistsn ε such that
whenever n, m ≥ n ε , if follows that |a n −a m | 0 be given and suppose a n → a. Then from the definition of convergence,
there exists n ε such that if n>n ε , it follows that |a n −a| < ε 2 . Therefore, if m, n ≥ n ε +1,
it follows that
|a n −a m |≤|a n −a| + |a − a m | < ε 2 + ε 2 = ε
showing that, since ε>0 is arbitrary, {a n } is a Cauchy sequence. It remains to that the
last claim.
Suppose then that {a n } is a Cauchy sequence and a = lim k→∞ a nk where {a nk } ∞ k=1
is a subsequence. Let ε>0 be given. Then there exists K such that if k, l ≥ K, then
k=1