Does this example prove the sequence is non-decreasing or non-increasing?

Conditions

• The sequence is defined by $a_{n+1}=\frac{1}{3}(a_n+1)$ with $a_1=1$.
• The lower bound $a_n \ge \frac{1}{2}$ has been proven.

Reasoning, step by step

1. Compute the difference: $a_{n+1}-a_n = -\frac{2}{3}a_n + \frac{1}{3}$.
2. Use the bound $a_n \ge \frac{1}{2}$ to show $-\frac{2}{3}a_n + \frac{1}{3} \le 0$.
3. Conclude $a_{n+1}-a_n \le 0$, which means $a_{n+1} \le a_n$.
4. Identify this condition as the definition of a non-increasing sequence.
5. Note the discrepancy between the spoken "non-decreasing" and the written "non-increasing".

Example

The board writes $a_{n+1}-a_n\le0\Rightarrow a_{n+1}\le a_n$ and later explicitly states "$\{a_n\}_{n=1}^\infty$ bounded and non-increasing".

Common misconceptions

• Relying solely on the spoken audio without checking the written mathematical inequalities.
• Confusing the direction of the inequality $a_{n+1} \le a_n$ with non-decreasing behavior.

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