(a) Given a>-1, a ≠0, use induction to show that (1+a)^n>1+n...

(a) Given $a>-1, a \neq 0$, use induction to show that $(1+a)^n>1+n a$ for any integer $n>1$. (b) Use (a) to show that, for any $x>0$, the sequence $(1+(x / n))^n$ increases. (c) If $a>0$, show that $(1+a)^r>1+r a$ holds for any rational exponent $r>1$. [Hint: If $r=p / q$, then apply (a) with $n=q$ and (b) with $x=a p$.] (d) Finally, show that (c) holds for any real exponent $r>1$.

(a) Given $a>-1, a \neq 0$, use induction to show that $(1+a)^n>1+n a$ for any integer $n>1$.
(b) Use (a) to show that, for any $x>0$, the sequence $(1+(x / n))^n$ increases.
(c) If $a>0$, show that $(1+a)^r>1+r a$ holds for any rational exponent $r>1$.
[Hint: If $r=p / q$, then apply (a) with $n=q$ and (b) with $x=a p$.]
(d) Finally, show that (c) holds for any real exponent $r>1$.

Key Concepts

Density of Rationals and Extension to Real Exponents

The density of rational numbers in the real numbers means that every real number can be approximated arbitrarily closely by a rational number. Using this property, results proven for rational exponents can be extended to real exponents by approximating a real exponent with a sequence of rational exponents. This concept, often combined with the continuity properties of exponential functions, is essential for generalizing inequalities from rationals to the full set of real numbers.

Monotonic Sequences

A monotonic sequence is one that is either entirely non-decreasing or non-increasing. In proving that a sequence like (1+(x/n))^n is increasing, the concept of monotonicity is pivotal. Monotonic sequences often have properties that aid in establishing limits and convergence, and showing monotonic behavior is a key step in many analysis proofs.

Exponential Functions and Properties of Exponents

Exponential functions involve expressions where a constant base is raised to a variable exponent. The properties of exponents, such as the relationship between (1+a)^n and linear expressions like 1+na, are central in many algebraic inequalities. Understanding these properties is crucial when extending results from integer to rational and real exponents and when handling inequalities involving exponentiation.

Mathematical Induction

Mathematical induction is a fundamental proof technique used to prove that a statement holds for all natural numbers. It consists of verifying the base case and then proving that if the statement holds for some arbitrary integer, it also holds for the next integer. This technique is essential in establishing properties that depend on an integer parameter, such as recursive inequalities or definitions.

Algebraic Inequalities

Algebraic inequalities compare expressions and establish that one expression is greater than or less than another. In this context, proving that (1+a)^n > 1+na and its extensions relies on manipulating and comparing algebraic structures. Mastery of inequalities is important in many areas of mathematics, allowing for the estimation and bounding of functions and sequences.

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