Where is the Monotone Convergence Theorem used in this recursive-sequence proof?

Conditions

• The sequence has been proven to be bounded from below.
• The sequence has been proven to be non-increasing.

Reasoning, step by step

1. Verify the sequence is bounded below by $\frac{1}{2}$.
2. Verify the sequence is non-increasing ($a_{n+1} \le a_n$).
3. Apply the Monotone Convergence Theorem to conclude $a_n \to L$ for some real number $L$.
4. Use the existence of $L$ to pass to the limit in the recurrence relation.

Example

The board writes “$\{a_n\}_{n=1}^\infty$ bounded and non-increasing” and then “Monotone Convergence $\Rightarrow a_n\to L$.”

Common misconceptions

• Trying to find the value of the limit before proving that the limit exists.
• Assuming the theorem provides the value of the limit directly rather than just its existence.

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