Question about series rearrangement in Baby Rudin (theorem 3.54).

First, let me make sure it's clear what's happening behind all those formulas. The rearrangement that allegedly works goes like this: First take just enough positive terms from your given series to produce a partial sum $>\beta$. (You can do that because the series of all the positive terms diverges.) After that, put just enough negative terms to bring the partial sum down below $\alpha$ (possible because the series of all the negative terms diverges). Then resume putting just enough positive terms to bring the partial sum back up above $\beta$. Continue working back and forth like this.

Notice that I said just enough terms at every stage. That ensures that, when you get a partial sum $s$ above $\beta$, it won't be too far above $\beta$; the difference $s-\beta$ will be at most the last term you added, because otherwise you could have stopped adding positive terms sooner. Similarly, when the partial sum goes below $\alpha$, the difference will be (in absolute value) at most the (absolute value of the) last term you added.

But your original series converged (conditionally), so the terms approach zero. That means that the amounts by which you overshoot $\beta$ and undershoot $\alpha$ are eventually arbitrarily small as you perform more and more stages of the process. And that's what those last two lines in Rudin's proof are saying is "clear".