Please provide proofs for parts i.)-iii.)
(i) Refer to the sequence in 1(ii). Show that with respect tothe supremum norm on ?[0,1] this is a bounded sequence that has noconvergent subsequence. (hint: What is the value of ‖?? − ??‖∞ if ?≠??)
(ii) Refer to the sequence in 1(v). Show that this is a boundedsequence with respect to the 1-norm on ?[0,1] that has noconvergent subsequence.
(iii) Let â„Ž?(?) = sin??. Show that with respect to the 2-norm?[0,2?], (â„Ž?) is a bounded sequence that has no convergentsubsequence. (This exercise shows that the Bolzano-WeierstrassTheorem does not generalise to ?[?,?] with any of the 3 “naturalâ€norms on ?[?,?])
Note: sequences from 1ii.) and 1v.) are pointwise functions andare defined respectively below:
1ii.) For ? ≥ 2, define the function ?? on [0,1] by: ??(?) =(??, if 0 ≤ ? ≤ 1/?)
(2-??, if 1/?< ? ≤ 2/n)
(0, if 2/?< ? ≤ 1)
1v.) Hn=n?? (and fn is defined as above)
