18.100B Problem Set 4

Due in class Monday, March 11. You may discuss the problems with other students, but you should write solutions entirely on your own.

1. Suppose {p_{n}} and {q_{n}} are two sequences in a metric space X, that {p_{n}} converges to p, and that {q_{n}} converges to q. Prove that

lim_{n→∞} d(p_{n}, q_{n}) = d(p, q).

2. Suppose that {x_{n}} is a sequence of non-negative real numbers converging to a non-negative real number x. Prove that

lim_{n→∞} \sqrt{x_{n}} = \sqrt{x}.

(Hint: it may help to consider separately the cases x = 0 and x ≠ 0.)

3. Find a sequence of real numbers for which every positive integer is a subsequential limit.

4. This problem concerns the 2-adic metric on ℤ, defined by

d_{2}(m, n) = \{ \begin{matrix}0, & if m = n \\ 2^{−a}, & if m ≠ n and 2^{a} is the largest power of 2 dividing m − n. \end{matrix}

Let X be the metric space ℤ with the 2-adic distance. Show that the sequence {1, 3, 7, 15, 31, … } in X converges. (This can be phrased in terms of infinite series as saying that the series

1 + 2 + 4 + 8 + ⋯ = ∑_{j=0}^{∞} 2^{j}

converges in X. You may know a formula for summing such a series that will help you guess the limit of the sequence.)

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