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Once convergence to a limit $L$ is established, pass to the limit in the recurrence relation $a_{n+1}=\frac{1}{3}(a_n+1)$. Since both $a_n$ and $a_{n+1}$ approach $L$, substitute $L$ for both terms to get $L=\frac{1}{3}(L+1)$, and then solve this linear equation for $L$.

Conditions: The sequence is known to converge to some real number $L$.; The recurrence relation is $a_{n+1}=\frac{1}{3}(a_n+1)$.

Once convergence to a limit $L$ is established, pass to the limit in the recurrence relation $a_{n+1}=\frac{1}{3}(a_n+1)$. Since both $a_n$ and $a_{n+1}$ approach $L$, substitute $L$ for both terms to get $L=\frac{1}{3}(L+1)$, and then solve this linear equation for $L$.

Conditions: The sequence is known to converge to some real number $L$.; The recurrence relation is $a_{n+1}=\frac{1}{3}(a_n+1)$.