I am learning about limits for the first time and while reading old questions on here, I stumbled upon this: Question about sums of limit superiors
Could someone clarify a step for me?
We have set $N = \max\{N_1,N_2\}$ and got $s_n+t_n < \alpha + \beta + \epsilon$. So far everthing is clear to me. However, then they follow from this that:
$\sup \{s_n+t_n|n>N\} \leq \alpha + \beta + \epsilon$ and therefore $\limsup (s_n+t_n) \leq \alpha + \beta$.
I don't understand how we get from $s_n+t_n < \alpha + \beta + \epsilon$ to $\sup \{s_n+t_n|n>N\} \leq \alpha + \beta + \epsilon$ and why from this we can conclude the limit superior as smaller or equal to $\alpha + \beta$.