Why does the lecturer say the sequence must be bounded and monotone to use the Monotone Convergence Theorem?

Conditions

• The sequence consists of real numbers.
• The goal is to prove convergence using the Monotone Convergence Theorem.

Reasoning, step by step

1. Identify the theorem being applied: the Monotone Convergence Theorem.
2. Recall the hypotheses of the theorem: the sequence must be bounded and monotone.
3. Conclude that verifying these two properties is the required proof strategy.

Example

The lecturer states they want to use the monotone convergence theorem and therefore need to show the sequence is bounded and non-decreasing or non-increasing.

Common misconceptions

• Believing that boundedness alone is sufficient for convergence.
• Assuming that monotonicity alone is sufficient for convergence.

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