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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.
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.
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 $x_n=n$.
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...
No.
A subsequence is a sequence taken from the original, where terms are selected in the order they appear.
For example, let $x_n = \frac{1}{n}$. Let's take a subsequence $x_{n_j}$ where we pick every other term, i.e.
$$ x_{n_j} = \left(1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \ldots \right) $$
Notice how these terms in the subsequence are taken in the order they appear.
Consider a sequence $$a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8, \dots$$
Intuitively, we can create a subsequence by selecting an infinite amount of random terms of this sequence, and throwing what's left away. However, it is important that if we select $a_i$ and $a_j$ with $i <j$, then also $i < j$ in the subsequence. In other words: we keep the order of the elements in the original sequence.
For example, the sequence:
$$a_1,a_3,a_4,a_7, \dots$$ would be a subsequence of the sequence written earlier.
In this case, we can then find a strictly increasing function $k: \mathbb{N} \to \mathbb{N}$ such that $$1 \mapsto 1$$ $$2 \mapsto 3$$ $$3 \mapsto 4$$ $$4 \mapsto 7$$ $$\dots$$
and write $b_n = a_{k_{n}}$ to denote the subsequence .
The strictly increasing part is important to preserve the order of the elements as in the original sequence.
This leads us to the following definition:
A sequence $(y_n)_n$ is subsequence of a function $(x_n)_n$ iff there exists a strictly increasing function $k: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$, we have $y_n = x_{k(n)}$
you should not ask whether you can take just one element from the sequence once or not as $(1,1,1,1.....)$ is surely is not a subsequence of $(1,2,3,4,5.....)$ but $(1,1,1,1,1,1.....)$ is subsequence of $(-1,1,-1,1,-1,1.....)$. Hence you should consider your subsequence as if your original sequence is say $\{x(n)\}_{n=1}^\infty$ then your subsequence will be anything of the form $\{x(n_i)\}_{i=1}^\infty$ for some $n_1<n_2<n_3.....$.
