Analysis I. Sheet 2. Question 3(iii).
I think that sup(f(S)) = f(sup(S)) should be sup(f(S)) <= f(sup(S)).
For example, define the increasing function f: R --> R by:
f(x) = x, for all x < 1.
f(x) = x + 1, for all x >= 1.
Take S to be the open interval (0,1).
Then f(S) = (0,1), so sup(f(S)) sup((0,1)) = 1.
But sup(S) = 1, so f(sup(S)) = f(1) = 2.
Analysis I. Sheet 4. Question 3(iii).
The second occurrence of u_n should be w_n.
Analysis I : Sheet 7 Question 3
I apologise for two typos :
(i) In part (a) both sums should start at 0.
(ii) In part (c) the exponent of (1-x) is -(k+1) and not -k.