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The Monotone Convergence Theorem states that every bounded, monotone sequence of real numbers converges. Therefore, to prove convergence using this theorem, one must establish both boundedness and monotonicity as the necessary hypotheses.

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

The Monotone Convergence Theorem states that every bounded, monotone sequence of real numbers converges. Therefore, to prove convergence using this theorem, one must establish both boundedness and monotonicity as the necessary hypotheses.

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

The proof uses mathematical induction. First, the base case is verified by checking that $a_1=1 \ge \frac{1}{2}$.

Conditions: The sequence is defined by $a_{n+1}=\frac{1}{3}(a_n+1)$ for $n \ge 1$.; The initial value is $a_1=1$.; The goal is to prove $a_n \ge \frac{1}{2}$ for all $n \ge 1$.

The proof uses mathematical induction. First, the base case is verified by checking that $a_1=1 \ge \frac{1}{2}$.

Conditions: The sequence is defined by $a_{n+1}=\frac{1}{3}(a_n+1)$ for $n \ge 1$.; The initial value is $a_1=1$.; The goal is to prove $a_n \ge \frac{1}{2}$ for all $n \ge 1$.