Monotone Convergence Theorem (Real Analysis)/Decreasing Sequence

Theorem

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

  • 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Exercise $4$
  • 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Theorem $1.2.6$
  • 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
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