# Monotone Convergence Theorem (Real Analysis)

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

## Theorem

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.