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Subtract $a_n$ from both sides of the recurrence relation. Distribute the $\frac{1}{3}$ and combine like terms to simplify the expression to $-\frac{2}{3}a_n + \frac{1}{3}$.

Conditions: The recurrence relation is $a_{n+1}=\frac{1}{3}(a_n+1)$.; The goal is to analyze the sign of the difference to determine monotonicity.

Subtract $a_n$ from both sides of the recurrence relation. Distribute the $\frac{1}{3}$ and combine like terms to simplify the expression to $-\frac{2}{3}a_n + \frac{1}{3}$.

Conditions: The recurrence relation is $a_{n+1}=\frac{1}{3}(a_n+1)$.; The goal is to analyze the sign of the difference to determine monotonicity.