4
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$ Jun 22, 2020 at 7:58

2 Answers 2

4
$\begingroup$

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 $n_k=k^2$.

$\endgroup$
4
$\begingroup$

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 $(x_{n_i})_{i\in \mathbb N}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.