Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example.

A monotonic sequence, including the definitions of increasing sequences, decreasing sequences, strictly increasing sequences, strictly decreasing sequences, eventually increasing sequences, and eventually decreasing sequences

The given statement is that a monotonic sequence, including the definitions of increasing sequences, decreasing sequences, strictly increasing sequences, strictly decreasing sequences, eventually increasing sequences, and eventually decreasing sequences.

• A sequence $a_n$is monotonic increasing if $a_{n+1}\ge a_n\mathrm{for}\mathrm{all}n\in N$
  
• The sequence isstrictly monotonic increasing if we have ">" in the definition and similarly for Monotonic decreasing sequences as they have "<" in the sequence.
• The elements of the sequences $\frac{1}{n},1-\frac{1}{n}\mathrm{and}1+\frac{1}{{10}^n}$ seem to approach a single point as n increases. In these sequences the values are either increasing or decreasing as n increases, but they “eventually approach” a single point. Though the elements of the sequence $\frac{{\left(-1\right)}^n}{n}$oscillate, they eventually approach the single point 0.
• For example, a sequence by$a_{n+1}=\sqrt{2a_n-1}\mathrm{with}{a}_1=2$is a monotonic decreasing sequence.