Question

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: Every sequence is a function.

(b) True or False: The third term of the sequence \(\left \{ k+1 \right \}_{k=1}^{\infty }\) is \(4\).

(c) True or False: The third term of the sequence \(\left \{ k^2 \right \}_{k=2}^{\infty }\) is \(9\).

(d) True or False: Every sequence of real numbers is either increasing or decreasing.

(e) True or False: Every sequence of numbers has a smallest term.

(f) True or False: Every recursively defined sequence has an infinite number of distinct outputs.

(g) True or False: Every sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

(h) True or False: Every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

Short answer

Part a. The statement is True.

Part b. The statement is True.

Part c. The statement is False.

Part d. The statement is False.

Part e. The statement is False.

Part f. The statement is False.

Part g. The statement is False.

Part h. The statement is True.

Step-by-step solution

Part (a) Step 1. Given Information

We are given a statement 'Every sequence is a function'.

The objective is to know whether the statement is true or false.

Part (a) Step 2. Explanation

Every sequence is a function from the positive integer to the real numbers.

Hence, a sequence is a function that has a set of natural numbers as its domain.

So the given statement is True.

Part (b) Step 1. Given Information

We are given a statement '

The third term of the sequence \(\left \{ k+1 \right \}_{k=1}^{\infty }\) is \(4\)'.

The objective is to know whether the statement is true or false.

Part (b) Step 2. Explanation

To find the third term of the sequence substitute \(k=3\) in \(k+1\). So,

$\begin{array}{rcl}k+1& =& 3+1\\ & =& 4\end{array}$

So the third term of the sequence \(\left \{ k+1 \right \}_{k=1}^{\infty }\) is \(4\)'.

Hence, the given statement is True.

Part (c) Step 1. Given Information

We are given a statement 'The third term of the sequence \(\left \{ k^2 \right \}_{k=2}^{\infty }\) is \(9\)'.

The objective is to know whether the statement is true or false.

Part (c) Step 2. Explanation

To find the third term of the sequence substitute \(k=4\) in \(k^2\). So,

$\begin{array}{rcl}{k}^2& =& 4^2\\ & =& 16\end{array}$

So the third term of the sequence \(\left \{ k^2 \right \}_{k=2}^{\infty }\) is \(16\)'.

Hence, the given statement is False.

Part (d) Step 1. Given Information

We are given a statement 'Every sequence of real numbers is either increasing or decreasing'.

The objective is to know whether the statement is true or false.

Part (d) Step 2. Explanation

Consider the following sequence \(\left \{ 1,-1,1,-1,1,-1 \right \}\).

This sequence is neither increasing nor decreasing.

So the given statement is false.

Part (e) Step 1. Given Information

We are given a statement 'Every sequence of numbers has a smallest term'.

The objective is to know whether the statement is true or false.

Part (e) Step 2. Explanation

A sequence which is infinitely decreasing may not have a least term. As it value can go as low as negative infinity.
So it is not necessary that every sequence of numbers has a smallest term.

Hence the given statement is false.

Part (f) Step 1. Given Information

We are given a statement 'Every recursively defined sequence has an infinite number of distinct outputs'.

The objective is to know whether the statement is true or false.

Part (f) Step 2. Explanation

Consider a recursively defined sequence \(a_{k}=a_{k-1}\) and \(a_0=1\). For this sequence for \(k=1\).

$\begin{array}{rcl}{a}_1& =& a_{1-1}\\ & =& a_0\\ & =& 1\end{array}$

The above defined sequence is a constant sequence which each term equal to \(1\).

So recursively defined sequence may not have an infinite number of distinct outputs.

Hence the given statement is false.

Part (g) Step 1. Given Information

We are given a statement 'Every sequence has an upper bound, a lower bound, or both an upper bound and a lower bound'.

The objective is to know whether the statement is true or false.

Part (g) Step 2. Explanation

Consider the sequence \(\left \{ \left ( -1 \right )^{k} k\right \}_{k=0}^{\infty }\).

The terms of the sequence are \(\left \{ 0,-1,2,-3,... \right \}\).

This sequence is neither bounded above nor below.

Hence, the given statement is false.

Part (h) Step 1. Given Information

We are given a statement 'Every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound'.

The objective is to know whether the statement is true or false.

Part (h) Step 2. Explanation

Every monotonic sequence is either an increasing sequence or a decreasing sequence.

Therefore it has an upper bound, or a lower bound, or both an upper bound and a lower bound.

Hence, the given statement is true.