The sequence is defined by and . Prove by induction on that for all and hence show that as .
Let , , and for , let .
(a) Show that
Hence, using the induction hypothesis, :
show that for all .
(b) Use (a) above and an additional induction on to show that
for all .
(c) Since as (from elsewhere in the webpages) we have
Use this together with (b) to prove directly from the definition of convergence for that as .
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.
; ; ; .