Monotone Convergence Theorem (Real Analysis)/Examples/n - 1 over n

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:

When $n = 1$ we have:

So for all $n \in \R_{>0}$ we have that $a_{n + 1} - a_n > 0$.

Thus $\sequence {a_n}$ is strictly increasing.

The result follows from the Monotone Convergence Theorem (Real Analysis).