The statement that \(q_1, q_2, q_3, \dots\) is a subsequence of \(p_1, p_2, p_3, \dots\) means that there is an increasing sequence of natural numbers, \(n_1, n_2, n_3, \dots\) such that for each natural number \(i,\) we have \(p_{n_i} = q_i\text{.}\)
Example. Suppose \(p_1, p_2, p_3, \dots\) is a sequence and \(n\) is a function with domain the natural numbers defined by \(n(k) = 2k\text{.}\) Then \(n\) defines the subsequence: \(p_2, p_4, p_6, \dots\text{.}\)
Suppose that \(q_1, q_2, q_3, \dots\) is a subsequence of \(p_1, p_2, p_3, \dots\text{.}\) Show that if there is a number \(x\) so that \(p_1, p_2, p_3, \dots\) converges to \(x\) then \(q_1, q_2, q_3, \dots\) converges to \(x\text{.}\)
Suppose that \(q_1, q_2, q_3, \dots\) is a subsequence of \(p_1, p_2, p_3, \dots\) and there is a number \(x\) so that \(q_1, q_2, q_3, \dots\) converges to \(x\text{.}\) Is it true that \(p_1, p_2, p_3, \dots\) converges to \(x\text{?}\)
Suppose that \((p_n)_{n=1}^{\infty}\) is a sequence of points in the closed interval \([a,b]\text{.}\) Is it true that every subsequence of \((p_n)_{n=1}^\infty\) converges to some point in \([a,b]?\)
A set of numbers \(K\) is compact if every sequence of points in \(K\) has a subsequence that converges to some point in \(K\text{.}\)
Show that every closed interval is compact.
If \(x\) is a limit point of \(\{p_1, p_2, p_3, \dots \}\) and every subsequence of \((p_n)_{n=1}^\infty\) converges then \((p_n)_{n=1}^\infty\) converges to \(x\text{.}\)
Show that every closed and bounded set in \(\mathbb{R}\) is compact.
Previously, we proved that every infinite bounded set has a limit point. Now we have the equivalent to this statement for sequences, that every sequence with infinite bounded range has a convergent subsequence.
The statement that the sequence \(p_1,p_2,p_3, \dots\) is a Cauchy sequence means that if \(\epsilon\) is a positive number, then there is a positive integer \(N\) such that if \(n\) is a positive integer and \(m\) is a positive integer, \(n \ge N\text{,}\) and \(m\ge N\text{,}\) then the distance from \(p_n\) to \(p_m\) is less than \(\epsilon\text{.}\)
The sequence \(p_1, p_2, p_3, \dots\) is a Cauchy sequence if and only if it is true that for each positive number \(\epsilon\text{,}\) there is a positive integer \(N\) such that if \(n\) is a positive integer and \(n \ge N\text{,}\) then \(|p_n-p_N| \lt \epsilon\text{.}\)
If the sequence \(p_1,p_2,p_3,\dots\) converges to a point \(x\text{,}\) then \(p_1,p_2,p_3,\dots\) is a Cauchy sequence.
If \(p_1,p_2,p_3,\dots\) is a Cauchy sequence, then the set \(\{p_1,p_2,p_3,\dots\}\) is bounded.
If \(p_1,p_2,p_3,\dots\) is a Cauchy sequence, then the set \(\{p_1,p_2,p_3,\dots\}\) does not have two limit points.
If \(p_1,p_2,p_3,\dots\) is a Cauchy sequence, then the sequence \(p_1,p_2,p_3,\dots\) converges to some point.