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25 Answer the questions of qn 24 in relation to the two sequences
1, 0,
,0,
,0,
,0,...,
,0,...
and
0, 0, 0, 0, . . ., 0, . . ..
The similarity of answers in qns 24 and 25 will make us accept the
sequences in qn 25 as 'tending to 0'.
26 Exhibit the inadequacy of the following attempts (a)
—
(e) to define a
null sequence, that is, a sequence which tends to 0, by reference to
the succeeding set of sequences (i)
—
(v), noneof which is null.
(a) A sequence in which each term is strictly less than its
predecessor.
(b) A sequence in which each term is strictly less than its
predecessor while remaining positive.
(c) A sequence in which, for suMciently large
n
, each term is less
than somesmall positivenumbr.
(d) A sequence in which, for suMciently large
n
, theabsolute
value of each term is less than some small positive number.
(e) A sequence with arbitrarily small terms.
(i) 2, 1, 0,
1,
2,
3,
4, . . .,
n
,...
(ii) 2,
,...
(iii) 2, 1, 0,
0.1,
0.1,
0.1, . . .,
0.1, . . .
(iv) 2, 1, 0,
0.1, 0.01,
0.001, 0.01,
0.001, . . ., 0.01,
0.001, . . .
(v) 1,
,
,
,
,...,
,1,
,1,
,...
27 Examine the sequence
1,
,
,
,
,...,(
1)
/
n
,...
(i) Beyond what stage in the sequence are the remaining terms
between
0.1 and 0.1?
(ii) Beyond what stage in the sequence are the remaining terms
between
0.01 and 0.01?
(iii) Beyond what stage in the sequence are the remaining terms
between
0.001 and 0.001?
(iv) Beyond what stage in the sequence are the remaining terms
between
and
, where
is a given positive number? The
choiceof
, the Greek
e
, here, is to stand for 'error', where the
terms of the sequence are thought of as successive attempts to
hit thetargt 0.
