Monotone Convergence Theorem (Real Analysis)/Examples/Power of Real Number between Zero and One
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Example of Use of Monotone Convergence Theorem (Real Analysis)
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 $0$.
Proof
From Power of Real Number between Zero and One is Bounded, $\sequence {a_n}$ is bounded below with supremum $0$.
As $x < 1$, it follows from Real Number Ordering is Compatible with Multiplication that:
- $x^{k + 1} < x^k$
The result follows from the Monotone Convergence Theorem (Real Analysis).
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.18$: Examples: $\text{(ii)}$