Monotone Convergence Theorem (Real Analysis)/Decreasing Sequence

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Theorem

Let $\sequence {x_n}$ be decreasing and bounded below.


Then $\sequence {x_n}$ converges to its infimum.


Proof

Let $\sequence {x_n}$ be decreasing and bounded below.

Then $\sequence {-x_n}e$ is increasing and bounded above.

Thus the Monotone Convergence Theorem for Increasing Sequence applies and the proof follows.

$\blacksquare$


Sources