Q. Explain exactly what it means for $\{a_n\}$ $n\in\mathbb N$ to converge to $L ∈ R.$
I wrote that for $\{a_n\}$ to converge to $L ∈ R$ means that the infinite sequence $\{a_n\}$ has a limit $L$ where no matter how small $ε > 0$ you choose, from a certain $n$ onwards, the absolute difference $|L - \{a_n\}|$ between $L$ and $\{a_n\}$ is always less than ε.
Am I on the right track? Is there anything else that needs to be added?
"$\{a_n\}$ converges to $L \in \mathbb R$" and "$\{a_n\}$ has the limit $L \in \mathbb R$" mean the same thing. The point of convergence is called the limit, so to say a sequence is convergent is the same thing as saying it has a limit. However, as you go further down the chapter, you will learn about "limit points", and this notion will clash slightly with the notion of limit, because saying "$\{a_n\}$ has a limit $a$" and "$\{a_n\}$ has a limit point $a$" will mean two different things. For this reason, I think you should stick with "$a_n$ converges to $a$" rather than "$a_n$ has limit $a$" in writing/speech.
The definition is actually very fine. However, when you write $|L - \{a_n\}|$ you must be careful : the notation $\{\cdot\}$ denotes a sequence , and when we refer to a term of that sequence, then we remove the bracket. A sequence(of real numbers) is a function from the natural numbers to the real numbers : a term of the sequence is a single real number. We don't know how to subtract a sequence from a real number : we know how to subtract a real number though. Therefore, $|L-a_n|$ is the right notation to be used there.
Besides this, I find your definition to be suitable.