Exercise.
The function :
is
defined by ()=1
2+1
.
(a) Find
such that
()
11000
=
.
(b) Show that
()
11000000
1000000
.
(c) Given that
means
interpret the statement in (b) above using ∀ and
.
(d) Let
0
be an arbitrary positive
real number. Show that there is some
such that
()
.
Exercise.
The following define seqences (
)
which
all converge to 0.
In each case, do the following.
(a) Find
such that
1100
. (Note that you do not have to give the best
value
, but you must prove all assertions you make.)
(b) Do the same for 11000 ...
You may use only the following facts about the function
:
and the function
10:(0,)
:
-
()
1
for all
and
-
10 is an increasing function with
10(10)
= for all
.
Possible open-ended discussion for a tutorial: which of these sequences
approaches 0 fastest? What does this mean? and how would you prove it?