What is a subsequence in calculus?

For example, if I have the sequence $(1,2,3,4,5,6,7,8,\ldots)$ i.e $x(n) = n$ for all natural numbers, then is the subsequence $(1,1,1,1,1...)$ valid? Or can I only take one element from the sequence once? Would the subsequence $(1,2,1,2,1,2...)$ be a valid subsequence?

Thanks.

Solution 1:

In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements

Formally, a subsequence of the sequence $(a_n)_{n \in \mathbb{N}}$ is any sequence of the form $(a_{n_k})_{k \in \mathbb{N}}$ where $(n_k)_{k \in \mathbb{N}}$ is a strictly increasing sequence of positive integers.

Hence, your two examples are not valid, as $1$ appears exactly once in the original sequence.

However if the original sequence is $(1,2,3,4,5,6,7,8,1,2,3,4,5,6,7,8,\ldots)$ that is the $8$ numbers are periodic, then yes, it is a valid subsequence.

Solution 2:

A sequence of real numbers is a map $x:\Bbb N\to\Bbb R$; we typically write it as $n\mapsto x_n$ (rather than $n\mapsto x(n)$).

A subsequence of this sequence is a map $y=x\circ\phi$ where $\phi:\Bbb N\to \Bbb N$ is a strictly increasing map. Then $y_n=x_{\phi(n)}$. Neither of your examples is a subsequence of your given sequence which has $x_n=n$.

Solution 3:

This may be easier to visualize: Write your sequence (any sequence) $$7,8,5,6,4,4,5,7,7,7,7,8,7,3,2,6,1,2,6,2,8,3,3,2,3,\ldots$$ Then select some elements $$7,\underline{8},5,6,\underline{4},4,5,\underline{7},7,7,7,8,\underline{7},3,\underline{2},6,\underline{1},\underline{2},6,\underline{2},8,3,\underline{3},2,3,\ldots$$ and erase the rest $$8,4,7,7,2,1,2,2,3,\ldots$$ That is a subsequence...