I am very confused at everything to do with subsequences. The author defines a subsequence as the following:
Given a sequence:
$(a_n) = (1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, ...)$
let $n_1 \lt n_2 \lt n_3 \lt n_4 \lt ...$ be an increasing sequence of natural numbers.
Then the sequence
$a_{n_1}, a_{n_2}, a_{n_3}, a_{n_4}, ...$ is called a subsequence of $(a_n)$ and is denoted by $(a_{n_j})$ with $j \in \mathbb{N}$.
All fine and dandy. A subsequence is a sequence composed of elements of another sequence in the same order, without repeats. It is basically "dropping" all indexes but the chosen $j's$.
The author now wishes for me to prove that
Subsequences of a convergent sequence converge to the same limit as the original sequence.
What I know is that is a convergent sequence is bounded, so intuitively this bound $M \gt 0$ will be the limit of $a_n$. I started a bit of a proof and immediately got lost. Lets say I have a sequence:
$\{1, 2, 3, 4, 5, 6, 7\}$.
Intuitively, the limit of this sequence is $7$. But if I take any subsequence:
$\{2, 4, 6\}$
It's limit is $6$, is it not? How can this subsequence's limit be $7$?
I don't want help with the proof as much as help clearing up this really bad misunderstanding of subsequences. I would prefer to struggle with the proof with the intuition so that I can fully grasp it.
My though was that since a sequence can have many upper bounds (for example, $7, 8, 9, 10, ...$ all are upper bounds for my example sequence) this would lead to the intuition. Another thought I had was subsequences in the question referred to the set of all subsequences of the convergent sequence, in which case we would be talking about something like a limit superior which would make more sense. However, I am hopelessly lost.
Can anyone explain to me the intuition that leads one to recognize a subsequence's limit is the limit of its "parent sequence"?