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MAT 237 Notes on 1.5 Subsequences

In section 1.5 a recurring theme is the passage from a sequence to a subsequence. “ ... for any sequence
{xk:k= 1,2, . . . }... there is a subsequence {xkj:j= 1,2, . . . }which converges ... "
A sequence can be looked at as a garden which is not very well kept. There may be a very strong design
which may be hidden under lots of unwanted herbs and weed. If we get rid of the weed then we can see the
strong design that the gardener had in mind. This is the process of passing to a subsequence, that is, to get
rid of the unessential terms of the original sequence so that we can showcase the most interesting part.
The notation for subsequences may be somewhat confusing, and it is the purpose of this note to elaborate
on the notation of subsequence.
Given a sequence {x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11 . . . }one can extract a subsequence {xkj:j=
1,2. . . }arbitrary as follows:
{x3, x4, x7, x11, x25, x76, x78, x1034, . . . }
. In this case the ﬁrst term of the subsequence, indexed by k1is x3, that is the index k1= 3. Similarly the
second term (naturally indexed by k2,) is the term x4, that is k2= 4,and similarly k3= 7, k4= 11, k5=
25, k6= 76, k7= 78, k8= 1034 etc.
Important convention: The subsequnce index kjis to be considered as an increasing function of
j, which assigns a kvalue to any input of j. That is k1is the ﬁrst index among the indices kselected.
Increasing means that the values assigned to the increasing values of jwil have to be strictly increasin. That
is k1< k2< k3< ... (see top of page 27 for this convention.) This convention becomes very useful in the
proof of theorem 1.20 almost the last line of the proof.
In the proof of theorem 1.19 this process of reducing to a subsequence is repeated several time (exactly
ntimes.) So let’s try to repeat this process twice, that is let’s reduce the above subsequence once more to
another subsequence. Say the new subsequence of the old subsequence is
{x3, x11, x78, . . . }
In this case, the new indexes must be be doubly indexed to refer to the process of selection from a subsequence.
That is, the new subsequence is {xkji:i= 1,2,3. . .. In this case we can see that the ﬁrst term of the new
subsequnce, for i= 1 is 3, that is xkj1=x3or that the index kj1=k1= 3, or that j1= 1 and similarly
kj2=k4= 11 (that is j2= 4), while kj3=k7= 78 (that is j3= 7.)
Another important technique in working with sequences is that they are already indexed by natural num
bers. In arguments relevant to subsequences we are often interested in making a selection (of subsequences)
with certain properies. But in general we are faced with lots of terms which satisfy our criterion, which one
should we select? A natural selection would be the term with smallest index which satisﬁes the criterion.
This idea can further clarify the proof of 1.18)
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