Using Induction to Prove Sequence Decrease

Use induction to show that the sequence $y_1, y_2, y_3, \ldots$ is decreasing, i.e., $y_1 \geq y_2 \geq y_3 \geq \cdots$.

In this problem, we are tasked with using the principle of mathematical induction to demonstrate that a given sequence is decreasing. Induction is a powerful proof technique often used in mathematical reasoning to establish a statement for all natural numbers. It consists of two major steps: the base case and the inductive step. The base case involves proving that the statement holds for the initial value, often n equals one. Once the base case is verified, the inductive step involves assuming the statement holds true for an arbitrary natural number n, and then proving it also holds for n plus one. This creates a domino effect, allowing the statement to hold for all natural numbers.

A decreasing sequence in real analysis is a sequence where each term is greater than or equal to the term that follows it. Ensuring that a sequence decreases involves showing that $y_n \geq y_{n+1}$ for all n. Importantly, you must express each term with respect to its predecessor in a way that supports this inequality. Sometimes, knowing the particular form or relationship between consecutive terms can help in establishing this inequality clearly.

When attempting this problem, it may be helpful to examine the sequence's general form or any additional properties that can simplify establishing the required inequality. Such properties might include monotonicity or the boundedness of the sequence, depending on how $y_n$ is defined. Understanding these aspects will not only allow you to apply induction correctly but also deepen your comprehension of how sequences can behave and be characterized in real analysis.