I'm on Jech Chapter 3 (Cardinal Numbers) on the section on cofinalities. I don't understand why the implication in the red rectangle is true. If the aforementioned gamma-sequence was constant and every term equaled to alpha, why would the right hand limit in the red rectangle necessarily evaluate to gamma?
The key is the phrase "in $\alpha$" - that is, $\beta_\eta<\alpha$ for each $\eta<\gamma$. This rules out the possibility that each $\beta$ is equal to $\alpha$, and in fact means that while the sequence of $\beta$s is not necessarily increasing at each step (it's only guaranteed to be nondecreasing) it does grow unboundedly-in-$\gamma$-often.