18.1.3. Universality of the Statement

Try these first

1. 1.

   What is a subsequence?

   Hint

   Think of sequences like (1, 3, 2). Which numbers can form a subsequence?

2. 2.

   Define a strictly increasing sequence.

   Hint

   Consider the sequence (a, b, c) — what should a be compared to b?

3. 3.

   What is a strictly increasing subsequence?

   • Elements can be equal
   • Each element is less than the previous
   • Each element is greater than the previous

   Hint

   Think of an example where this applies.

4. 4.

   True or False - Every sequence of 5 distinct real numbers contains a subsequence of length 5 that is either increasing or decreasing.

   • True
   • False

   Hint

   Reflect on the definition of subsequences.

5. 5.

   Prove the statement with a specific example of seven distinct real numbers.

   Hint

   Arrange numbers visually to help identify patterns.

6. 6.

   Demonstrate the pigeonhole principle clearly with 10 numbers and 3 distinct categories (such as odd, even).

   Hint

   Digging into the counts will help clarify.