18.1.1. Definition of Increasing and Decreasing Sequences

Try these first

1. 1.

   Define a strictly increasing sequence.

   Hint

   Think of a sequence that rises steadily.

2. 2.

   What is a subsequence?

   Hint

   Look for an example in the sequence.

3. 3.

   What is a strictly increasing sequence?

   • A sequence where all elements are the same
   • A sequence where each element is less than the previous one
   • A sequence where each element is greater than the previous one

   Hint

   Look at how the terms relate to one another.

4. 4.

   True or False: Every sequence has at least one increasing subsequence.

   • True
   • False

   Hint

   Consider different orders of numbers.

5. 5.

   Given the sequence of numbers from 1 to 20, demonstrate and find at least three different subsequences that are strictly increasing.

   Hint

   Look for numbers that increase without skipping downward.

6. 6.

   Construct a sequence of your choice of distinct real numbers and show that it contains both an increasing and decreasing subsequence.

   Hint

   Select numbers that have both upward and downward movements.