VI. Sequences #

Problem 1 #

Suppose that \((S, \rho)\) is a metric space. If \(T\) is a subset of \(S\) and \(t\) is a limit point of the set \(T\), show that there is a sequence of points \(\{p_n\}_{n=1}^\infty\) in \(T\) such that

$$\lim p_n = t.$$

Solutions: Hint · Solution · Long solution

Problem 2 #

Consider a sequence \(\{s_n\}\) of real numbers satisfying \(a \leq s_n \leq b\) for some \(a,b \in \mathbb{R}\) and all \(n \in \mathbb{N}\). Suppose that the sequence converges and that \(\lim s_n = s\). Show that \(a \leq s \leq b\).

Solutions: Hint · Solution · Long solution

Problem 3 #

There are several equivalent definitions of the notion of a bounded sequence in \((\mathbb{R},|\cdot|)\). The following is often the easiest one to work with:

Definition. \(\{c_n\}\) is a bounded sequence of real numbers iff there is a positive number \(D \in \mathbb{R}\) such that \(|c_n| \leq D\) for all \(n \in \mathbb{N}\).

So let \((S,\rho) = (\mathbb{R}, |\cdot|)\) and consider a sequence \(\{s_n\}\) such that \(\lim s_n = 0\). Suppose further that \(\{c_n\}\) is a bounded sequence. Show that the sequence of their products \(\{c_n s_n\}\) converges and that

$$\lim c_n s_n = 0.$$

Note that we have not assumed that \(\{c_n\}\) converges!

Solutions: Hint · Solution · Long solution

Problem 4 #

In our proof of the convergence of the alternating harmonic series, we will use the fact that for every integer \(m\) greater than \(n \in \mathbb{N}\), the following inequality holds

$$\Big|\frac{1}{n+1} - \frac{1}{n+2} + \frac{1}{n+3} - \ldots \pm \frac{1}{m}\Big| < \frac{1}{n}.$$

Prove it.

Solutions: Hint · Solution · Long solution

Problem 5 #

Define a sequence of real numbers \(s_n = \left(1 + \frac{1}{n}\right)^n\). Show that the sequence is monotone and bounded, concluding that it must converge. Its limit is defined as the familiar number:

$$\lim \left(1 + \frac{1}{n}\right)^n = e.$$

Feel free to use without proof the fact that terms of this sequence are always smaller than 3.

Hint: Use the binomial theorem. There are several ways to show that the sequence is monotone, but one builds on the observation that \(\binom{n}{2} \cdot \left(\tfrac{1}{n}\right)^2 = \tfrac{1}{2}\left(1-\tfrac{1}{n}\right)\).

Solutions: Hint · Solution · Long solution