what is a subsequence?

Question

i m getting confused in this subsequence According to definition of a subsequence a subsequence should be a strictly increasing sequence of number and also we can form a subsequence by deleting some terms and not changing the position of remaining terms s={1,1/2,1/4,1/8.....} is sequence a its subsequence can be a={1/2,1/8....} but this subsequence is not strictly increasing but it is decreasing then how can it be a subsequence?????

Answer 1

I think you're confusing terms with indices. If we have a sequence $$ a_1, a_2, a_3, \ldots $$ then no matter which numbers those actually are, we have that $$ a_1, a_3, a_5,\ldots $$ is a subsequence. In fact, for any increasing sequence $n_1, n_2, n_3\ldots$ of natural numbers, (the example above had $n_1 = 1, n_2 = 3, n_3 = 5,\ldots$, which is increasing), the sequence $$ a_{n_1}, a_{n_2}, a_{n_3},\ldots $$ is a subsequence of $a_1, a_2, \ldots$, regardless of whether $a_{n_1}, a_{n_2}, a_{n_3},\ldots$ itself is increasing.

Answer 2

A sequence with elements in a set $X$ is a function $a: \mathbb N\to X$. Traditionally we write $a_n:=a(n)$ and $(a_n):=a$.

Now consider any increasing function $f:\mathbb N\to \mathbb N$. A subsequence of $(a_n)$ is defined as a function $a\circ f:\mathbb N\to X$. From this definition, we see clearly that every subsequence is a sequence.

Example: Let $X=\mathbb R$ and $a:\mathbb N\to \mathbb R$ is $a_n=\frac 1n$ and $f:\mathbb N\to\mathbb N$ is $f(n)=2n$. Then if $b=a\circ f$ we have $b_n=\frac{1}{2n}$. Writing it explicitly:

$$(a_n):\frac {1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4},\dots$$ $$(b_n): \frac 12, \frac 14, ...$$

We see that $(b_n)$ is constructed by removing some elements of $(a_n)$.