Convergence of a Recursively Defined Sequence

Show that the sequence defined by $a_1 = 1$ and $a_{n+1} = 3 - \frac{1}{a_n}$ is increasing and $a_n < 3$ for all $n$. Deduce that the sequence $a_n$ is convergent and find its limit.

This problem focuses on analyzing the behavior of a recursively defined sequence. To approach this, it is important to understand the foundational concept of sequences in real analysis, particularly in terms of monotonicity and boundedness. Identifying whether a sequence is increasing or decreasing often involves examining its recursive or direct formula and leveraging mathematical induction or properties of the function governing transitions between successive terms.

For this problem, showing that the sequence is increasing requires demonstrating that each subsequent term follows a strict order greater than the previous. This can involve comparing the terms in a rigorous grammatical proof structure, analyzing the function involved in the sequence definition.

The next part is to establish an upper bound for the sequence. Showing that each term of the sequence stays below a particular value (in this case, 3) involves strategic examination of the recursive definition and utilizing algebraic manipulation or known inequalities. Such bounding behavior is critical as it ties directly into the concept of convergence in sequences.

The convergence of a sequence typically involves proving that it is both bounded and monotonic, by invoking the Monotone Convergence Theorem. The theorem is fundamental in real analysis and offers a precise pathway from recognizing a sequence’s structure to asserting its convergence. Lastly, finding the limit of the sequence entails solving an equation derived from setting subsequent terms equal in the context of their limit behavior, thereby translating recursive sequence behaviors into functional or numerical understanding.