Question

Beweisen Sie durch vollständige Induktion, dass die folgende Verallgemeinerung der BERNOULLIschen Ungleichung gilt: Für \(01-\left(a_{1}+a_{2}+\cdots+a_{n}\right) $$.

Short answer

The inequality holds for all integers \(n \geq 2\) by mathematical induction.

Step-by-step solution

Induction Basis

We start by checking if the statement holds for the base case, which is the smallest possible value of \(n\), typically \(n = 2\). For \(n = 2\), we need to prove that \((1-a_1)(1-a_2) > 1-(a_1+a_2)\). Expanding the left side gives \(1 - a_1 - a_2 + a_1a_2\). This simplifies to \(1 - (a_1 + a_2 - a_1a_2)\). Since \(a_1, a_2 < 1\), we have \(a_1a_2 < a_1\) and \(a_1a_2 < a_2\), so \(a_1a_2 < a_1 + a_2\). Thus, \(a_1a_2 - a_1 - a_2 < 0\), making \((1-a_1)(1-a_2) > 1-(a_1+a_2)\). The base case holds.

Induction Hypothesis

Assume the inequality holds for some arbitrary \(n = k\). That is, assume \((1-a_1)(1-a_2)\cdots(1-a_k) > 1-(a_1 + a_2 + \cdots + a_k)\). This assumption is our induction hypothesis.

Induction Step

We need to prove that if the inequality holds for \(n = k\), it also holds for \(n = k+1\). Consider \((1-a_1)(1-a_2)\cdots(1-a_k)(1-a_{k+1})\). By our induction hypothesis, we know \((1-a_1)(1-a_2)\cdots(1-a_k) > 1-(a_1 + a_2 + \cdots + a_k)\). Multiply both sides by \((1-a_{k+1})\), giving \[(1-a_1)(1-a_2)\cdots(1-a_k)(1-a_{k+1}) > (1-(a_1 + a_2 + \cdots + a_k))(1-a_{k+1})\]. Expanding the right side gives \(1 - (a_1 + \cdots + a_k) - a_{k+1} + (a_1 + \cdots + a_k)a_{k+1}\). Similar reasoning shows \(a_1 + \cdots + a_k + a_{k+1} > (a_1 + \cdots + a_k)a_{k+1}\). Therefore, \(1 - ((a_1 + \cdots + a_k) + a_{k+1}) < 1 - ((a_1 + \cdots + a_k) + a_{k+1}) + (a_1 + \cdots + a_k)a_{k+1}\) holds, proving \(n = k+1\) holds true.

Final Conclusion

Since the induction basis holds and the induction step shows the statement is true for \(n = k+1\) given it's true for \(n = k\), by mathematical induction, the inequality holds for all integers \(n \geq 2\).

Key concepts

Mathematical Induction

Mathematical induction is an essential technique used to prove a wide array of mathematical statements, especially those involving natural numbers. The approach involves two primary steps—the base case and the induction step. The base case confirms that the statement holds for the first natural number in the sequence, usually starting at 1 or 2. In our original exercise, the base case checks the smallest possible value for \(n\), which is 2. The statement was verified to be correct for \(n = 2\).
The next step is the induction step. Here, you assume the statement holds for an arbitrary natural number \(k\) (induction hypothesis) and then prove it's true for \(k+1\). Consequently, once both the base case and the induction step are shown to be true, the induction principle concludes that the statement is valid for all natural numbers following the base case. This method not only simplifies proof techniques but also guarantees that mathematical assertions maintain their validity across indefinitely large sets of numbers.

• The Base Case: Validate the statement for the starting number.
• The Induction Hypothesis: Assume the statement true for \(n = k\).
• The Induction Step: Prove the statement for \(n = k+1\).

Proof Techniques

Proof techniques form the backbone of validating mathematical arguments, ensuring they stand on rigorous and logical grounds. The method of proof by induction, used in the original exercise to establish Bernoulli's inequality, is a well-organized technique suitable for demonstrating truths about infinite sequences.
This process, while systematic, may require other types of proof techniques such as direct proofs, where statements are demonstrated logically from known facts, or contrapositive proofs, which involve proving that if the consequence is false, the premise must also be false. Moreover, understanding each component of the proof, as highlighted in the induction method, requires meticulous expansion and simplification of expressions, anticipating inequalities, and relying on established truths or hypotheses (like \(a_k < 1\) constraints in Bernoulli's inequality proof). Overall, proof techniques should be applied accurately to generate sound and convincing proofs.

• Direct Proofs: Logical progression from hypothesis to conclusion.
• Contrapositive Proofs: Prove false implication leads to false premise.
• Proof by Contradiction: Assume the opposite and derive a contradiction.

Inequalities

Inequalities play a significant role in mathematics, providing insights into the relative sizes of values within equations. Bernoulli's inequality, a classic example in advanced mathematics, shows that certain exponential expressions can be approximated or bounded by linear combinations. This exercise involves a generalized form of the inequality, suggesting that the product of backward expanded terms \((1-a_1)(1-a_2)...(1-a_n)\) is greater than the linear subtraction series \(1-(a_1+a_2+...+a_n)\).
Key to working with inequalities is understanding algebraic manipulations. In the step-by-step solution, the inequality is simplified by expanding and reconfiguring terms, subsequently applying conditions such as \(a_i < 1\) to draw conclusions. This complements the induction hypothesis and provides the necessary proof for every increasing \(n\). Proper handling and simplification of inequalities using these principles guide us to desired outcomes in calculus, number theory, and beyond.

• Understand the directional signs (greater than, less than).
• Perform algebraic manipulations accurately.
• Apply specific conditions or known inequalities.

Advanced Mathematics

Advanced mathematics encompasses complex concepts and detailed proofs, which are foundational for rigorous academic study and research. Tackling Bernoulli's inequality through induction is an instance of applying advanced mathematical techniques to contemplate deeper problems.
Understanding Bernoulli's inequality itself requires basic knowledge in algebra, inequalities, and sequences or sums. Higher-level mathematics involves extending these principles to broader and more abstract frameworks, which often require specialized techniques and proofs, like those evident in calculus and abstract algebra.
Working with these advanced topics means:

• Analyzing complex problems beyond basic math frameworks.
• Engaging with proofs to attain robust, logical conclusions.
• Continuously studying mathematical theories and applications in-depth.

An endeavor into advanced mathematics prepares students for challenges in research or data-driven fields, where problems are often open-ended and solutions less straightforward.