Is there any algebraic way to represent a subsequence converging to L? Let the subsequence be denoted by $(b_{n})$, then does this imply for all $n\geq N, \epsilon \gt 0 $, we have that: $\lvert b_{n} - L \rvert \lt \epsilon$ ? Of course each $b_{n}$ is in $(a_{n})$ correct?
This question is just about sequences in general and about the intuition behind them. I'm just a bit confused by the definition of subsequence and what it really means for a sequence to converge.
Thanks
On
Let's do an example. Here is a sequence.
$$ 0, 1, 0, 1, 0, 1, 0, 1, \ldots$$
Does this sequence have a limit? No, definitely not. But it has convergent subsequences. The sequence formed of the odd indexed terms (i.e. the 1st term, the 3rd term, the 5th, etc.) is the sequence $0, 0, 0, 0, \ldots$, which definitely converges to $0$. The sequence formed from the even index terms is the sequence $1,1,1,1, \ldots$, which definitely converges to $1$.
This is an example of a sequence with two different subsequences converging to two different values.
A sequence is a function on the set of positive integers. Let $a$ be a sequence, $f$ be a non-decreasing sequence of positive integers. Then the composition $a\circ f$ is called a subsequence of $a$. The convergence of a subsequence is therefore defined in the same way as the convergence of any sequence.