# Monotone Convergence Theorem (Real Analysis)

*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

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $\S 1.2$: Real Sequences: Theorem $1.2.6$

- 1953: Walter Rudin:
*Principles of Mathematical Analysis*... (previous): $3.14$