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The Monotone Convergence Theorem states that if a sequence $(a_n)_{n\in\mathbb{N}}$ is monotonically decreasing (i.e., $a_{n+1} \le a_n$ for all $n$) and bounded from below, then the sequence is convergent. This is a direct application of the existence of the infimum for bounded sets.

Conditions: The sequence $(a_n)$ is monotonically decreasing.; The sequence $(a_n)$ is bounded from below.

The Monotone Convergence Theorem states that if a sequence $(a_n)_{n\in\mathbb{N}}$ is monotonically decreasing (i.e., $a_{n+1} \le a_n$ for all $n$) and bounded from below, then the sequence is convergent. This is a direct application of the existence of the infimum for bounded sets.

Conditions: The sequence $(a_n)$ is monotonically decreasing.; The sequence $(a_n)$ is bounded from below.

The example proves the sequence is non-increasing. The mathematical derivation shows $a_{n+1}-a_n \le 0$, which implies $a_{n+1} \le a_n$.

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.

The example proves the sequence is non-increasing. The mathematical derivation shows $a_{n+1}-a_n \le 0$, which implies $a_{n+1} \le a_n$.

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.