Question

Please provide proofs for parts i.)-iii.)

(i) Refer to the sequence in 1(ii). Show that with respect to the supremum norm on ?[0,1] this is a bounded sequence that has no convergent subsequence. (hint: What is the value of ‖?? − ??‖∞ if ? ≠ ??)

(ii) Refer to the sequence in 1(v). Show that this is a bounded sequence with respect to the 1-norm on ?[0,1] that has no convergent subsequence.

(iii) Let ℎ?(?) = sin??. Show that with respect to the 2-norm ?[0,2?], (ℎ?) is a bounded sequence that has no convergent subsequence. (This exercise shows that the Bolzano-Weierstrass Theorem does not generalise to ?[?,?] with any of the 3 “natural” norms on ?[?,?])

Note: sequences from 1ii.) and 1v.) are pointwise functions and are 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)

Answer 1