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.

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.