# Monotone Convergence Theorem (Real Analysis)/Decreasing Sequence

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## Theorem

Let $\sequence {x_n}$ be decreasing and 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.

$\blacksquare$

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.5$: Limits: Exercise $4$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $\S 1.2$: Real Sequences: Theorem $1.2.6$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.17 \ \text{(ii)}$ - 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces

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