Monotone Convergence Theorem (Real Analysis)/Increasing Sequence
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Theorem
Let $\sequence {x_n}$ be an increasing real sequence which is bounded above.
Then $\sequence {x_n}$ converges to its supremum.
Proof
Suppose $\sequence {x_n}$ is increasing and bounded above.
By the Continuum Property, it has a supremum, $B$.
We need to show that $x_n \to B$ as $n \to \infty$.
Let $\epsilon \in \R_{>0}$.
By the definition of supremum, $B - \epsilon$ is not an upper bound.
Thus:
- $\exists N \in \N: x_N > B - \epsilon$
But $\sequence {x_n}$ is increasing.
Hence:
- $\forall n > N: x_n \ge x_N > B - \epsilon$
But $B$ is still an upper bound for $\sequence {x_n}$.
Then:
\(\ds \forall n > N: \, \) | \(\ds B - \epsilon\) | \(<\) | \(\ds x_n \le B\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall n > N: \, \) | \(\ds B - \epsilon\) | \(<\) | \(\ds x_n < B + \epsilon\) | Real Plus Epsilon | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall n > N: \, \) | \(\ds \size {x_n - B}\) | \(<\) | \(\ds \epsilon\) | Negative of Absolute Value: Corollary 1 |
Hence the result.
$\blacksquare$
Sources
- 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous): $3.14$
- 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: $\S 4.17 \ \text{(i)}$
- 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