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  3. please provide proofs for parts i.)-iii.) (i) refe

Please provide proofs for parts i.)-iii.) (i) Refer to the sequence in 1(ii). Show that with...

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Advance Math

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 ℎ?(?) = sin??. Show that with respect to the 2-norm?[0,2?], (ℎ?) is a bounded sequence that has no convergentsubsequence. (This exercise shows that the Bolzano-WeierstrassTheorem does not generalise to ?[?,?] with any of the 3 “natural”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)

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