Monotone Convergence Theorem (Real Analysis)/Decreasing Sequence

Theorem
Let $\sequence {x_n}$ be a decreasing real sequence which is bounded below.

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

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

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

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