Understanding the proof of every convergent sequence of real numbers is bounded.

Question

I want to understand why in this proof:

https://proofwiki.org/wiki/Convergent_Real_Sequence_is_Bounded

It was taken $\epsilon = 1$. Why was it needed to specify a value for $\epsilon$? Wouldn't the proof be true if I just take the bounds as $K= \max\{|x_1|,|x_2|,...,|x_N|,|L|+\epsilon\}$?

Answer 1

Only once you make $K$ specific - by specifying $\epsilon$. But you could have taken $\epsilon=42$ instead of $\epsilon=1$, it would not matter for the overall result.