Definition of subsequence used in defining accumulation points

Question

According to the definition given on this page, a number $a$ is an accumulation point of a sequence $(a_n)$ if there is a subsequence $(a_{n_k})$ that converges to $a$ in the $\lim_{k\to \infty}$.

What does the word subsequence mean in this definition?

In the page linked above, Theorem 1 says that if a sequence converges, then it has only one accumulation point, namely, the value that the original sequence converges to.

For example, example 2 on the page linked above claims that the only accumulation point of $$(a_n) : a_n = \frac{n+1}{n}, \quad n \in \mathbb{N}$$

is $1$ because $\lim_{n \to \infty} a_n = 1. $

Now, Wikipedia says that a subsequence is formed by deleting terms from the parent sequence. So, what if we picked the subsequence consisting only of the first term with $n=1$? Then $a_1 =2$ and $2$ appears to be an accumulation point.

Must the subsequence be infinite in size, and if so, how can we modify the Wikipedia definition of a subsequence to require that an infinite number of terms remain after deletion?

Answer 1

A subsequence of a sequence $(a_n)_{n\in\Bbb N}$ is a sequence $(a_{n_k})_{k\in\Bbb N}$ such that $(n_k)_{k\in\Bbb N}$ is a strictly increasing sequence of natural numbers. So, for instance, $(a_{2n})_{n\in\Bbb N}$ and $(a_{n^2})_{n\in\Bbb N}$ are subsequence of $(a_n)_{n\in\Bbb N}$. In the first case, I took $n_k=2k$ and, in the second case, I took $n_k=k^2$.

Answer 2

A formal way to define a subsequence is the following.

First let us recall what is the formal definition of a sequence. A sequence in a set $X$ is a function $f:\mathbb N\to X$. One may write $x_n$ to mean $f(n)$, and $f$ now may be denoted as $(x_n)_{n\in \mathbb N}$.

A strictly monotonically increasing function from $\mathbb N$ to $\mathbb N$ is a map $\phi:\mathbb N\to \mathbb N$ such that $\phi(i)<\phi(j)$ whenever $i<j$.

Now, if $f$ is a sequence in $X$, a subsequence of $f$ is any function of the form $f\circ \phi$, where $\phi:\mathbb N\to \mathbb N$ is a strictly mononotonically increasing function. If $(x_n)$ denotes the sequence $f$, and $\phi(i) = n_i$, then we may denote the seqeunce $f\circ \phi$ as $(x_{n_i})_{i\in \mathbb N}$