Monotone Convergence Theorem (Real Analysis)/Examples/n - 1 over n
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Example of Use of Monotone Convergence Theorem (Real Analysis)
The sequence $\sequence {a_n}_{n \mathop \ge 1}$ defined as:
- $a_n = \dfrac {n - 1} n$
is convergent to the limit $1$.
Proof
From Set of Numbers of form n - 1 over n is Bounded Above, $\sequence {a_n}$ is bounded above with supremum $1$.
Then we have that:
\(\ds \forall n \in \N_{\ge 2}: \, \) | \(\ds a_{n + 1} - a_n\) | \(=\) | \(\ds \dfrac {\paren {n + 1} - 1} {n + 1} - \dfrac {n - 1} n\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac n {n + 1} - \dfrac {n - 1} n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n^2 - \paren {n + 1} \paren {n - 1} } {\paren {n - 1} n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n^2 - \paren {n^2 - 1} } {\paren {n - 1} n}\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {\paren {n - 1} n}\) | simplifying | |||||||||||
\(\ds \) | \(>\) | \(\ds 0\) | as both $n > 0$ and $n - 1 > 0$ |
When $n = 1$ we have:
\(\ds a_{n + 1} - a_n\) | \(=\) | \(\ds \dfrac {\paren {1 + 1} - 1} {1 + 1} - \dfrac {1 - 1} 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 - \dfrac 0 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2\) | ||||||||||||
\(\ds \) | \(>\) | \(\ds 0\) |
So:
- $\forall n \in \N_{>0}: a_{n + 1} - a_n > 0$
Thus $\sequence {a_n}$ is strictly increasing.
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: $\text{(i)}$