Question

(a) Find a real number \(c\) and an \(N_{0} \in \mathbb{N}\) such that \(n^{2}

Short answer

(a) \( c = 1, N_{0} = 2 \); (b) \( c = 1, N_{0} = 3 \).

Step-by-step solution

Evaluate the Inequality Condition (a)

To solve part (a), we need to find real number \( c \) and a natural number \( N_{0} \) such that \( n^{2} < c n^{3} \). Simplify the given inequality to \( n^{2}/n^{3} < c \), which can be rewritten as \( 1/n < c \). Clearly, as \( n \rightarrow \, \infty \), \( 1/n \rightarrow \, 0 \), hence any positive \( c > 0 \) will satisfy this for sufficiently large \( n \).

Choose Values for c and N₀ (a)

Let's choose \( c = 1 \). This means we need \( 1/n < 1 \), which is true for any \( n > 1 \). Therefore, we can set \( N_{0} = 2 \), ensuring \( n \geq 2 \). Thus, the inequality \( n^{2} < n^{3} \) holds for all \( n \geq 2 \).

Evaluate the Inequality Condition (b)

For part (b), we find \( c \) and \( N_{0} \) such that \( 5n < c n^{3} \). Rearranging gives \( 5/n^{2} < c \). As \( n \rightarrow \, \infty \), \( 5/n^{2} \rightarrow \, 0 \), so choose \( c > 0 \) to satisfy this for large \( n \).

Choose Values for c and N₀ (b)

Choose \( c = 1 \), leading to \( 5/n^{2} < 1 \). We need \( n^{2} > 5 \). Checking, \( n = 3 \) gives \( n^{2} = 9 > 5 \), so \( N_{0} = 3 \), ensuring \( n \geq 3 \) satisfies \( 5n < n^{3} \).

Key concepts

Real Numbers

Real numbers are all the numbers you can think of. They combine both rational numbers (like fractions and integers) and irrational numbers (numbers that can't be expressed as simple fractions, like \( \pi \) or \( \sqrt{2} \)).
Real numbers are used in mathematics because they can represent a continuous line of numbers, which is important for explaining things like distance or time.
The real number system is very flexible!

• Any real number, positive or negative (or zero), can be used in equations and inequalities.
• Real numbers have important properties like being able to order them in size (e.g., -2.5 is less than 3.6).

In the context of the exercise, we look for a real number \( c \) that helps us frame inequalities.
This flexibility allows for a variety of solutions depending on the circumstances.

Natural Numbers

The natural numbers are the set of positive integers starting from 1, which are \( \{1, 2, 3, \ldots\} \). They are the starting point for counting and ordering.
Natural numbers are pivotal in defining and solving inequalities, like in our problem where we need to find an \( N_0 \) such that certain conditions hold for all \( n \geq N_0 \).

• Natural numbers do not include zero, negatives, or fractions.
• They are straightforward and help establish sized increments in number lines.

When we mention \( N_0 \), we refer to the smallest natural number for which our inequality holds consistently.
This concept simplifies our understanding of the minimum requirements for the solution to be valid.

Mathematical Proofs

Mathematical proofs are logical arguments that demonstrate the truth of a mathematical statement.
They often involve a series of steps, like our exercise, to reach a conclusion.
Proofs are crucial for validating results and ensuring the step-by-step accuracy that defines mathematics.

• Proofs involve assumptions, such as choosing \( c \) in our scenario.
• Proofs methodically verify whether conditions hold, such as checking values for \( n \) in our inequalities.

In this problem, we worked through algebraic manipulation by simplifying given inequalities to find values that satisfy the conditions.
This approach verifies that a specified \( c \) and \( N_0 \) exist so the inequality holds for large enough natural numbers \( n \).
The process illustrates how mathematical proofs substantiate ideas, ensuring that mathematical statements aren't just assumed, but are backed by reason.