Thread: examples, sequences of ordinals

Give examples of strictly increasing sequences of ordinals such that

$\displaystyle \text{lim}_n(\alpha_n + \beta) \not = \text{lim}_n \alpha_n + \beta$,

$\displaystyle \text{lim}_n(\alpha_n + \beta_n) \not = \text{lim}_n \alpha_n + \text{lim}_n \beta_n$.

I cannot think any examples that work for this one. The problem is that I can't recall many examples of strictly increasing sequences of ordinals. I need some help on this problem. Thank you.

The same question was asked (and answered) here
sequences of ordinals
and here
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