Monotone Convergence Theorem (Real Analysis)

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

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.


Sources