Question

Show every subsequence of a subsequence of a given sequence is itselfa subsequence of the given sequence. Hint: Define subsequences asin (3) of Definition 11.1 g

Answer 1

Answer: Suppose that $S_{n}$,  $K_{n}$ and $A_{n}$ are all sequences, also suppose that  $A_{n}$ ⊂ $K_{n}$ ⊂ $S_{n}$  ∀ $n$ ∈ $Z^{+}$. This implies that $A_{n}$ ⊂ $S_{n}$ as required. 

Step-by-step explanation:

If you suppose that $S_{n}$,  $K_{n}$ and $A_{n}$  are all representing three sequences respectively, and you also suppose that  $A_{n}$ is a subset (in other words subsequence) of $K_{n}$ and $K_{n}$  is a subset (in other words subsequence) of $S_{n}$, where n takes its values from the set of positive integers. We can safely say that $A_{n}$ is a subsequence of $S_{n}$ and this can demosnstrated mathematically as follows:

Suppose that $S_{n}$,  $K_{n}$ and $A_{n}$ are all sequences, also suppose that  $A_{n}$ ⊂ $K_{n}$ ⊂ $S_{n}$  ∀ $n$ ∈ $Z^{+}$. This implies that $A_{n}$ ⊂ $S_{n}$ as required.