Show the following statement from class: Let \(\left( X,d\right)\) be a metric space and let \(A\subseteq X\). Then \(A\) is closed if and only if any sequence of elements of \(A\) that converges has its limit in \(A\). In other words if \(x_n\in A\) for all \(n\in\mathbb N\) and \(\lim_{n\to\infty}x_n=x\) then \(x\in A\).
(Assignment sheet 2 problem 15) Show that if a sequence \(\{x_n\}\) converges in a discrete metric space, then it is eventually constant.
Find a metric space \(\left( X,d\right)\) and a set \(A\subseteq X\) such that \(A\) is closed and bounded but is not sequentially compact.
(Assignment sheet 4 problem 4) The real axis and the imaginary axis divide \(\mathbb C\) into four quadrants as follows:
c|c \(\Omega_2\) & \(\Omega_1\) \(\Omega_3\) & \(\Omega_4\)
Determine the images of \(\Omega_1, \Omega_2, \Omega_3, \Omega_4\) under the exponential map \(z\mapsto e^z\).
(Assignment sheet 4 problem 6(b)) Give examples to illustrate that, in general, for complex numbers \(z\), \(w\), \[Log (zw)\ne Log (z)+Log (w).\]