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Definition (The Image Of A Sequence Is A Set)

Given a sequence $(a_n)$ of real numbers, note that the image of the sequence, namely $a(\mathbb{N}) = \{a_n\mid n\in\mathbb{N}\}$, is a subset of the real numbers. Because the image of the sequence is a set of real numbers, we can use any of our previous words that we defined on sets of real numbers, and now apply them to a sequence. Here are some examples:

• We say a sequence is bounded if the image of the sequence is a bounded set.
• A lower bound for a sequence is a lower bound for the image of the sequence.
• The supremum of a sequence is the supremum of the image of the sequence.
• A limit point of a sequence is a limit point of the image of the sequence.

Problem 73: (Limits Of Sequences And Limit Points Of Images)

Is a limit point of the image of a sequence equal to the limit of that sequence?

• Give an example of a sequence $(a_n)$ that converges to $L$, such that $L$ is a limit point of the image of the sequence.
• Give an example of a sequence $(a_n)$ that converges to $L$, such that $L$ is not a limit point of the image of the sequence.
• Make a conjecture about when limits of sequences and limit points of images of sequences are equal.
• Give an example of a sequence $(a_n)$ that does not converge, yet the image of the sequence has one (or more) limit points.

Problem 75: (Proving A Rational Sequence Converges)

Consider the sequence $\ds (s_n)=\left(\frac{4n^3-2n^2-7n}{5n^3-3n^2+2n-1}\right)$. In this problem, your job is to prove that $s_n\to 4/5$. You may assume that for all natural numbers, we have $5n^3-3n^2+2n-1>0$.

1. Show that $\ds(s_n-4/5)= \left(\frac{2 n^2-43 n+4}{5 \left(5 n^3-3 n^2+2 n-1\right)}\right).$
2. Find a $k_1>0$ and $M_1$ so that $|2 n^2-43 n+4|\leq k_1 n^2$ for all natural numbers $n>M_1$.
3. Find a $k_2>0$ and $M_2$ so that $|5 \left(5 n^3-3 n^2+2 n-1\right)|\geq k_2 n^3$ for all natural numbers $n>M_2$.
4. Prove that $(s_n)$ converges to $4/5$.

Problem 76: (Limit Of Quotient Equals Quotient Of Limits)

Suppose $(a_n)$ converges to $A$ and $(b_n)$ converges to $B\neq 0$, and also suppose $b_n\neq 0$ for every natural number $n$. Prove that $(a_n/b_n)$ converges to $A/B$.

Definition (Increasing Decreasing Monotonic Sequences)

Let $(a_n)$ be a sequence of real numbers.

• We say that $(a_n)$ is (strictly) increasing if $a_n<a_{n+1}$ for every $n\in\mathbb{N}$.
• We say that $(a_n)$ is (strictly) decreasing if $a_n>a_{n+1}$ for every $n\in\mathbb{N}$.
• We say that $(a_n)$ is nonincreasing if $a_n\geq a_{n+1}$ for every $n\in\mathbb{N}$.
• We say that $(a_n)$ is nondecreasing if $a_n\leq a_{n+1}$ for every $n\in\mathbb{N}$.
• We say that $(a_n)$ is monotonic if $(a_n)$ is either nonincreasing or nondecreasing.

You should notice that a strictly decreasing sequence is nonincreasing, and a strictly increasing sequence is nondecreasing.

Problem 77: (Monotonic Sequences Converge If And Only If Bounded)

Let $(a_n)$ be a monotonic sequence. Prove that $(a_n)$ converges if and only if $(a_n)$ is bounded.

Definition (Diverges To Infinity)

Let $(a_n)$ be a sequence of real numbers. We say that $(a_n)$ diverges to infinity if for every $V$, there exists $H$ such that for every $n\in \mathbb{N}$ we know $n>H$ implies $a_n>V$. When $(a_n)$ diverges to infinity, we write $(a_n)\to \infty$.

Problem 79: (Diverges To Negative Infinity)

Construct a definition of what it means to diverge to $-\infty$, by appropriately modifying the definition of diverges to $\infty$. Then prove that the sequence $(-n^3+2n)$ diverges to $-\infty$.

For more problems, see AllProblems