Monotone Convergence Theorem (Real Analysis)/Examples/Power of Real Number between Zero and One

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: $\text{(ii)}$