Monotone Convergence Theorem (Real Analysis)/Decreasing Sequence

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

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

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

Then $\left \langle {-x_n} \right \rangle$ is increasing and bounded above.

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