Definition 58.
A sequence is a function from \(\N\) to \(\R\text{.}\)
Section 2.2 Sequences
Sometimes these are called point sequences to distinguish them from other potential sequences.
Even though technically the sequence is a function \(a\text{,}\) for any given \(i\in\N\) we will write \(a_i := a(i)\text{.}\) Then we may write a sequence \(a(1), a(2), a(3), \ldots\) as \(a_1, a_2, a_3, \ldots\text{;}\) or, more formally, as \(\set{a_i}_{i=1}^\infty\text{.}\) In order to keep the index visible (this is especially useful when we combine sequences), we will occasionally abuse notation slightly and write \(\set{a_i}_{i=1}^\infty\) when referring to the function or the image set \(\setof{a(i)}{i\in\N}\text{.}\)
Example 59.
Write down several sequences you are familiar with. If possible, give an algebraic formula for each \(a_i\) in terms of \(i\text{.}\)
There is a deep connection between sequences and accumulation points, which the next few problems will elucidate. First, a definition—one you may have seen in calculus in a different form.
Definition 60.
We say that the (point) sequence \(p = \set{p_i}_{i=1}^\infty\) converges to the point \(x\) if, given an open interval \(S\) containing \(x\text{,}\) there exist an \(N\in \N\) such that if \(n\ge N\) is also a positive integer then \(p_n \in S\text{.}\)
If a sequence does not converge to any point \(x\text{,}\) we say that it diverges, or is divergent.
If \(p\) converges to \(x\text{,}\) we often simply say that \(p\) converges, and write \(p \to x\text{.}\)
The first problem about this concept should be used as a place to test ideas for how to prove convergence. Make sure you remember all the axioms you've learned and facts you've proved about real numbers—you may need them!
Problem 61.
Consider the sequence given by \(p_n = \frac{1}{n}\) (remember, \(n\in \N\) is part of the definition of a sequence). Show that \(p = \set{p_i}_{i=1}^\infty\) converges to 0.
Example 62.
Before tackling any of the next few problems, try actually writing down the first 10-12 elements of each sequence.
Problem 63.
Consider the sequence given by \(p_n = 1 - \frac{1}{n}\text{.}\) Show that \(p\) converges to 1.
Problem 64.
Consider the sequence with even terms \(p_{2n} = \frac{1}{2n-1}\) and odd terms \(p_{2n-1} = \frac{1}{2n}\text{.}\) Show that \(p\) converges to 0.
Problem 65.
Consider the sequence with odd terms \(p_{2n-1} = \frac{1}{2n-1}\) and even terms \(p_{2n} = 1 + \frac{1}{2n}\text{.}\) Determine whether \(p\) converges to 0.
The following problem connects our two notions of accumulation points and limits of sequences. The most profound property of the real numbers is part of this connection, as we'll soon see.
Problem 66.
Show that if \(p\) converges to the point \(x\) and for each \(i\in\N\text{,}\) \(p_i \ne p_{i+1}\text{,}\) then \(x\) is an accumulation point of the image set of \(\set{p_i}_{i=1}^\infty\text{.}\)
Example 67.
Why do we need the restriction that \(p_i\ne p_{i+1}\text{?}\) Is this an absolutely necessary restriction for \(x\) to be an accumulation point of the image set?
Problem 68.
Show that the sequence \(\set{\frac{1}{i}}_{i=1}^{\infty}\) does not converge to a point other than zero.
Problem 69.
Show that if \(p\) converges to the point \(x\) and \(y\) is a point different from \(x\text{,}\) then \(p\) does not converge to \(y\text{.}\)
Basic familiar facts about sequence convergence are recalled next. In these proofs, you may have to think a little more explicitly about what the intervals around \(x\) look like in order to combine sequences. Try doing some examples with explicit numbers first to get a sense of how to approach them.
Problem 70.
Show that if \(c\) is a number and \(p = \set{p_i}_{i=1}^\infty\) converges to \(x\text{,}\) then \(cp\) (the product of \(c\) with each term of \(p\)) converges to \(cx\text{.}\)
Problem 71.
Show that if \(q=\set{q_i}_{i=1}^\infty\) converges to \(y\) and \(p = \set{p_i}_{i=1}^\infty\) converges to \(x\text{,}\) then \(\set{q_i + p_i}_{i=1}^\infty\) converges to \(y+x\text{.}\)
Products and quotients of sequences behave like you think they will as well, and you can use these facts in the rest of the problems and examples. We will include one special case soon.
