Monotone Convergence Theorem (Real Analysis)

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This proof is about convergence in real analysis. For other uses, see Monotone Convergence Theorem.


Every bounded monotone sequence is convergent.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.

Increasing Sequence

Let $\left \langle {x_n} \right \rangle$ be increasing and bounded above.

Then $\left \langle {x_n} \right \rangle$ converges to its supremum.

Decreasing Sequence

Let $\left \langle {x_n} \right \rangle$ be decreasing and bounded below.

Then $\left \langle {x_n} \right \rangle$ converges to its infimum.