Proving Monotonicity of a Recursive Sequence

Prove that the sequence $a_n$ of positive numbers defined by the recursive formula $a_1 = 1$ and $a_{n+1} = 3 - \frac{1}{a_n}$ is monotonically increasing, i.e., for all natural numbers $n$, $a_n < a_{n+1}$.

In this problem, you are tasked with proving that a recursively defined sequence is monotonically increasing. The main strategy in such proofs is to rely on mathematical induction or recursive reasoning. First, understand the recursive definition given: each term in the sequence is constructed based on the previous term. The sequence starts at a base case, which is often the initial term provided, here being one. The recursive formula in this problem provides a unique insight into the behavior of subsequent terms, driven by the operation of subtracting the reciprocal of the previous term from three.

When proving monotonicity, especially with a recursive sequence, the key steps involve assuming the property holds for an arbitrary term and then proving it holds for the subsequent term. This aligns with the principle of mathematical induction, a common technique for sequences. Here, you demonstrate that if the current term of the sequence satisfies the inequality relation, then the next term must inherently fulfill the same criteria, largely due to the properties of positive numbers and the mechanics of the operation in the recursive formula. Be mindful of the base case—its role ensures the induction step has a valid starting point, crucial for validating the entire proof. This problem exemplifies a critical skill in analyzing sequences and understanding their growth behavior in calculus and analysis courses.