Problem

Let X denote the subset of consisting of all sequences (x1, x2, …) such that ** converges...

Let X denote the subset of consisting of all sequences (x1, x2, …) such that ** converges. (You may assume the standard facts about infinite series. In case they are not familiar to you, we shall give them in Exercise of the next section.)

Show that if d is a metric for X, then

is a bounded metric that gives the topology of X. [Hint: If f(x) = x/(1 +x) for x > 0, use the mean-value theorem to show that f(a + b) - f (b) ≤ f(a).]

(a) Show that if , then converges. [Hint: Use (b) of Exercise to show that the partial sums are bounded.]
Show that the euclidean metric don is a metric as follows: If and , define
(a) Show that x • (y + z) = (x • y) + (x • z).
(b) Show that |x•y| ≤ ||x|| ||y||. [Hint: If , let a = 1/||x|| and b = 1/||y||, and use the fact that ||ax ±by|| ≥ 0.1
(c) Show that ||x + y|| ≤ ||x|| + ||y||. [Hint: Compute (x + y) (x + y) and apply (b).]
(d) Verify that d is a metric.
(b) Let . Show that if , then so are x + y and cx.
(c) Show that