A confusion(possible book mistake) about one of the proofs in Spivak's Calculus?

Question

In Chapter 5 - Function Limits, there is a proof that:

if

$|x - x_0| < 1; |x - x_0| < \frac{\epsilon}{2(|y_0| + 1)}; |y - y_0| < \frac{\epsilon}{2(|x_0| + 1)}$

then

$|xy = x_0y_0| < \epsilon$

The solution offered is:

Since $|x - x_0| < 1 => |x| < 1 |x_0|$

and when we transform $|xy - x_0y_0|$ we get:

$|xy - x_0y_0| \leq |x||y - y_0| + |y_0||x - x_0|$

and from here we can conclude:

$|x||y - y_0| + |y_0||x - x_0| < |x|\frac{\epsilon}{2(|x_0| + 1)} + |y_0|\frac{\epsilon}{2(|y_0| + 1)} < (1 + |x_0|)\frac{\epsilon}{2(|x_0| + 1)} + |y_0|\frac{\epsilon}{2(|y_0| + 1)}$

And the final proof as stated in the book(and the one that confuses me) is:

$(1 + |x_0|)\frac{\epsilon}{2(|x_0| + 1)} + |y_0|\frac{\epsilon}{2(|y_0| + 1)} = \frac{\epsilon}{2} + \frac{\epsilon}{2}$

It's obvious that $(1 + |x_0|)\frac{\epsilon}{2(|x_0| + 1)} = \frac{\epsilon}{2}$ , but what about the other one:

$|y_0|\frac{\epsilon}{2(|y_0| + 1)} = \frac{\epsilon}{2}$ I'm not sure why this one is true.How can $|y_0|$ be equal to $(|y_0| + 1)$?Is this a mistake or am I missing something?

Answer 1

It should be $|y_0|\frac{\epsilon}{2(|y_0| + 1)} < \frac{\epsilon}{2}$, because $\frac{|y_0|}{|y_0|+1}$ < 1. This will give: $$(1 + |x_0|)\frac{\epsilon}{2(|x_0| + 1)} + |y_0|\frac{\epsilon}{2(|y_0| + 1)} < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$$