TITLE: What is a subsequence in calculus?
QUESTION [14 upvotes]: 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.

REPLY [3 votes]: 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 $y_n = x_{k(n)}$