# Definition of subsequence used in defining accumulation points

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?

• Usually, a subsequence of an infinite sequence is also infinite: the fact that it has infinite many terms does not necessarily means that it has infinite many values. Jun 22, 2020 at 7:58

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 $$n_k=k^2$$.
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 $$i.
Now, if $$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}$$