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?
Best Answer
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 $n_k=k^2$.