A strictly monotonic increasing sequence is bounded below, then we can conclude that

A strictly monotonic increasing sequence is bounded below, then we can conclude thatans.

Let (an) and (bn) be bounded sequences, with bn̸ = 0 for all n. Consider thefour sequences(i) (an) (ii) (bn) (iii) (anbn) (iv) (an/bn)(a) Which of the sequences (i)–(iv) have convergent subsequences? Justifyyour answer. [6 marks](b) Suppose also that (an) and (bn) are monotonic. Which of the sequences(i)–(iv) must be convergent? Justify your answer.

The sequence x_n={1, 2, 3, 4, 5, 5, 6, ...} is .......... and ............. sequence.1 pointan increasing and unboundeda non-decreasing and unboundeda decreasing and unboundedan increasing and bounded

Let (an) be a sequence defined bya1 = 12, an+1 = 2 − 2an + 1 for n ≥ 1.(a) Show that an ∈ [0, 1] for all n. [4 marks](b) Show that (an) is increasing. [5 marks](c) Show that (an) is convergent, and calculate its limit.

Let (xn) be a bounded sequence in R. Show that there exist subsequences (xnk ) and(xmk ) of (xn) such thatlimk→∞ xnk = lim sup xn and limk→∞ xmk = lim inf xn.