Page 91 - DMTH401_REAL ANALYSIS

Unit 7: Convergent Sequence

In this case we also say that {p } converges to p or that p is the limit of {p } and we write pn  p Notes
n n
or lim limp = p
n
n¥
If {p } does not converge we say it diverges
n
If there is any ambiguity we say {p } converges/diverges in X
n
The set of all p is said to be the range of {p } (which may be infinite or finite). We say {p } is
n n n
bounded if the range is bounded.

Example: Notice that our definition of convergent depends not only on {p } but also
n
on X.
1
For example {1/1 : n  } converges in  and diverges in (0, ¥). Consider the following
sequence of complex number (i.e. X =  )
2
(a) If S = 1/n then limS = ; the range is infinite, and the sequence is bounded.
0
n n¥ n
(b) If S = n then the sequence {S } is divergent; the range is infinite, and the sequence is
2
n n
unbounded.
n
(c) If S = 1 + [(–1) /n] then the sequence {S } converges to 1, is bounded, and has infinite range.
n n
n
(d) If S = i the sequence {S } is divergent, is bounded and has finite range.
n n
(e) If S = 1 (n = 1, 2, 3, ...) then {S } converges to 1 is bounded.
n n
7.2 Properties of Convergent Sequences
Theorem:
(a) {p } converges to p  X if and only if every neighbourhood of p contains p for all but
n n
finitely many n.
(b) If p, p’  X and if {p } converges to p and to p’ then p = p’
n
(c) If {p } converges then {p } is bounded.
n n
(d) If E  X and if p is a limit point of E, then there is a sequence {p } in E such that p = limp
n n
n¥
t
Theorem: Suppose {S }, {t } are complex sequence with limS = S and limt = . Then
n n n¥ n n¥ n
(a) lim (S + t ) = S + t
n¥ n n
(b) lim C S = C  S and lim C + S = C + S for any number C.
n¥ n n¥ n
(c) lim S t = S
n¥ n n t
1 1
(d) lim =
n¥ S n S
Theorem:
k
(a) Suppose x   (n  ) and x = ( , n, . . .  ,n). Then {x } converges to x = ( , . . .,  ) if and
n n 1 k n 1 k
only if
lim  = (1  j  k)
n¥ j,n j

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