Properties of monotone sequences - exercises

Exercise.

Prove that the following define monotonic sequences.

(a) (1+1 ) 12 .

(b) +1 - 3 .

(c) 3 2-+2 .

(d) .

(e) the sequence ( ) where 0=0 and +1= + 2 2 .

Discuss the convergence of these sequences.

NOTE: The word discuss is used to frame deliberately open-ended questions. Say as much as you reasonably can in the time available. All your assertions should be proved. Discuss the convergence of these sequences clearly is asking for an indication of whether the sequences do or do not converge, and if possible to what.