proving limsup of example with Definition of Limit Superior from Royden

Question

$l$ is called limit superior of the sequence $\langle x_n\rangle$ if

1. given $\epsilon >0$, there exists $n$ such that $x_k<l+\epsilon$ for all $k\geq n$
2. given $\epsilon >0$ and given $n$ there exists $k\geq n$ such that $x_k>l-\epsilon$

Here is how I'm trying to understand it, for any arbitrary epsilon given, there are infinitely many terms of the sequence $x_k$ lies on the left side of $l+\epsilon$(can any of these terms be left of $l- \epsilon$? Im thinking yes, since there aren't any restriction on how far these $x_k$ can be on the left of $l$ but then it doesn't make any sense) where $k>n_0$, now at the same time if I look at any different $n$ I can always find some $x_{k_n}$ that is on the right side of $l-\epsilon$ (since the restriction of being on left of $l+\epsilon$ is only on $x_k$ I'm thinking these $x_{k_n}$ can go on the right side on the $l+\epsilon$). Now if we take $\epsilon$ to zero, I'm not even sure what is this $l$ means in the context.

looking at some examples $x_n=(-1)^n$, I see how $l=1$ satisfies condition 1. since, given epsilon, we can find some natural number $n_0$, and for all terms after $n_0$, they lies on the left side of $1+\epsilon$, essentially, in this case, any natural number would work, then for condition 2. since for each $n$ we have $x_k=1$ for all the even $k\geq n$ which is on right side of $1-\epsilon$. I believe my logic makes sense, but now I dont know how to write the proof of $\lim sup (-1)^n=1$ formally.

Answer 1

(1) is saying that any number larger than $l$ is an upper bound for all but finitely many points in the sequence, while (2) is saying any number less than $l$ is not an upper bound for infinitely many points in the sequence.

Your logic for the example $x_n = (-1)^n$ is perfect, all that is left to turn it into a proof is to write it in complete sentences. For example, the first half of the proof could say:

Let $\epsilon > 0$ and set $n=1$. Then for $k\ge n$, we either have $x_k = 1 < 1 + \epsilon$ or $x_k = -1 < 1+\epsilon$. This shows (1).

Can you try to write the second half yourself?