Exercise.
(a) Show that
+2
-
15=
17
-15
+
+1-15
Hence, using the induction hypothesis, ()
:
15
show that
15
for all .
(b) Use (a) above and an additional induction on to show that
0
-15
2
12
for all .
(c) Since
12
0
as
(from
elsewhere in the webpages) we have
0
12
Use this together with (b) to prove directly from the definition
of convergence for (
)
that
15
as
.
Exercise.
Either by quoting theorems on boundedness, subsequences, uniqueness of limits, or anything else from the course so far, or by arguing directly from the definition, prove that the following sequences do not converge to any limit.
=2
;
=(-1)
10000000+1
;
=
3
4
;
=
2
.