Bounded Real Sequence/Examples/1 minus Reciprocals

Example of Bounded Real Sequence

Let $\sequence {s_n}_{n \mathop \ge 1}$ be the real sequence defined as:

$s_n = 1 - \dfrac 1 n$

That is:

$\sequence {s_n}$ is the sequence of $1$ minus the reciprocals of the strictly positive integers.

Then $\sequence {s_n}$ is bounded.

Proof

We have that $n \ge 1$.

Hence:

$0 < \dfrac 1 n \le 1$

Hence:

$0 \le 1 - \dfrac 1 n < 1$

Hence:

$0$ is a lower bound of $\sequence {s_n}$

$1$ is an upper bound of $\sequence {s_n}$.

Hence the result by definition of bounded real sequence.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bounded sequence
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bounded sequence