What is the Monotone Convergence Theorem for decreasing sequences?

Conditions

• The sequence $(a_n)$ is monotonically decreasing.
• The sequence $(a_n)$ is bounded from below.

Reasoning, step by step

1. Verify that the sequence satisfies $a_{n+1} \le a_n$ for all $n$.
2. Verify that there exists a lower bound $L$ such that $a_n \ge L$ for all $n$.
3. Apply the Monotone Convergence Theorem to conclude that $(a_n)$ converges.
4. Identify the limit as the infimum of the set of sequence terms.

Example

The video writes the theorem on the board: 'If $(a_n)_{n \in \mathbb{N}}$ is monotonically decreasing and bounded from below, then $(a_n)_{n \in \mathbb{N}}$ is convergent,' presenting it as an important application of the completeness of real numbers.

Common misconceptions

• Assuming the theorem applies to any bounded sequence; the sequence must also be monotonic.
• Confusing the decreasing case with the increasing case; the increasing case requires the sequence to be bounded from above.

Watch the explanation

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