# real analysis – Understanding Rudin 3.3(d)

For reference, here is the full Theorem 3.3(d) from Rudin.

Suppose $${s_n}$$, $${t_n}$$ are complex sequences, and $$limlimits_{n to infty} s_n = s$$, $$limlimits_{n to infty} t_n = t$$. Then

(a) $$limlimits_{n to infty} left(s_n + t_nright) = s + t$$;

(b) $$limlimits_{n to infty} cs_n = cs$$, $$limlimits_{n to infty} left(c + s_nright) = c + s$$, for any number $$c$$;

(c) $$limlimits_{n to infty} s_n t_n = st$$;

(d) $$limlimits_{n to infty} frac{1}{s_n} = frac{1}{s}$$, provided $$s_n neq 0$$ ($$n = 1, 2, 3, ldots$$), and $$s neq 0$$.

There’s one key step in the proof of (d) that I don’t understand. First, Rudin uses convergence of $$s_n$$ to find an $$m in mathbb{N}$$ so that for all $$n geq m$$, we have $$|s_n – s| < frac{1}{s} |s|$$. He then asserts that for $$n geq m$$, we have
$$|s_n| > frac{1}{2} |s|.$$
This is a very important step, but I cannot follow it. I’ve tried contradiction and the triangle inequality, but I can’t get the inequality signs to line up. For example, I tried (for $$n geq m$$),
$$|s_n| = |(s_n – s) + s| leq |s_n – s| + |s| < frac{1}{2} |s| + |s| = frac{3}{2} |s|.$$
It seems as though I’ve bounded $$|s_n|$$ “in the opposite direction.” If I knew $$s$$ were positive, expanding the absolute values might work, but we only know it’s non-zero.

The rest of the proof looks pretty straightforward to me, with one slight doubt. He asserts the existence of an $$N$$ (I don’t know why he requires $$N > m$$ when he could just take the maximum of $$N$$ and $$m$$; does this make a difference?) so that $$n geq N$$ implies $$|s_n – s| < frac{1}{2} |s|^2 epsilon$$. As $$|s_n| > frac{1}{s} |s|$$, we have $$frac{1}{|s_n|} < frac{2}{|s|$$. We then have:
begin{align*} left lvert frac{1}{s_n} – frac{1}{s} right rvert & = left lvert frac{s – s_n}{s_n cdot s} right rvert \ & = frac{|s – s_n|}{|s||s_n|} \ & < frac{2|s – s_n|}{|s|^2} \ & < frac{2}{|s|^2} cdot frac{1}{2} |s|^2 epsilon \ & = epsilon end{align*}

I would appreciate some feedback on the above and some help with those two questions (how Rudin deduces $$|s_n| > frac{1}{2} |s|$$ and why he takes $$N > m$$ instead of $$max(N,m)$$.)

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