# Monotone Convergence Theorem (Real Analysis)

*This proof is about Monotone Convergence Theorem in the context of Real Analysis. For other uses, see Monotone Convergence Theorem.*

## Theorem

Let $\sequence {x_n}$ be a bounded monotone sequence sequence in $\R$.

Then $\sequence {x_n}$ is convergent.

### Increasing Sequence

Let $\sequence {x_n}$ be an increasing real sequence which is bounded above.

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

### Decreasing Sequence

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

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

### Graphical Illustration

The following diagram illustrates the Monotone Convergence Theorem:

## Examples

### Example: $\dfrac {n - 1} n$

The sequence $\sequence {a_n}_{n \mathop \ge 1}$ defined as:

- $a_n = \dfrac {n - 1} n$

is convergent to the limit $1$.

### Example: $x^n$ for $0 < x < 1$

Let $x \in \R$ such that $0 < x < 1$.

The sequence $\sequence {a_n}_{n \mathop \ge 1}$ defined as:

- $a_n = x^n$

is convergent to the limit $0$.

## Sources

- 1953: Walter Rudin:
*Principles of Mathematical Analysis*... (previous): $3.14$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Theorem $1.2.6$