Working directly from the definition, show that the sequences
= and =(-1). are not bounded,
but that =(-1) is bounded.
Exercise.
Let =1-12. Show
1 as .
Note that 1 for all
but the limit is not strictly less than 1.
Exercise.
Suppose () is a sequence that converges to some
. Suppose also that there
are 1,2 such that
12 for all ,
Show that 12.
Exercise.
Prove (and learn) the very useful Squeeze Rule that says that:
if sequences (), (), ()
satisfy
and both (), () converge to
the same limit , then
() also converges to this limit .