<p>
We cannot do something equivalent for the first term in <xref
ref="cauchyriemlimit"/>,
since now both <m>\D x</m> and <m>\D y</m> are involved,
and both change as <m>\D z\to0</m>.
Instead we apply the Mean-Value <xref ref="realmeanvalue"/> for
real functions,<fn>
We collect several theorems from calculus,
such as the Mean-Value Theorem for real-valued functions,
in the Appendix.
</fn> to the real and imaginary parts <m>u(z)</m> and
<m>v(z)</m> of <m>f(z)</m>.
<xref ref="realmeanvalue"/>
gives real numbers <m>0\lt a,b\lt 1</m> such that
<md>
<mrow>
\frac{u(x_0+\D x, \, y_0+\D y)-u(x_0+\D x, \, y_0)}{\D y} \amp
\ = \ u_y(x_0+\D x, \, y_0+a \, \D y)
</mrow>
<mrow>
\frac{v(x_0+\D x, \, y_0+\D y)-v(x_0+\D x, \, y_0)}{\D y} \amp
\ = \ v_y(x_0+\D x, \, y_0+b \, \D y) \,
</mrow>
</md>.
Thus
<md>
<mrow>
\amp \frac{f(z_0+\D x+ i \D y)-f(z_0+\D x)}{\D y} -
f_y(z_0)
</mrow>
<mrow>
\amp = \left( \frac{ u(x_0 + \D x, y_0 + \D y) - u(x_0
+ \D x, y_0) }{ \D y } - u_y(z_0) \right)
</mrow>
<mrow>
\amp \quad + i \left(
\frac{ v(x_0 + \D x, y_0 + \D y) - v(x_0 + \D x, y_0) }{
\D y } - v_y(z_0) \right)
</mrow>
<mrow xml:id="eq_laststepcauchyr" number="yes">
\amp = \left(u_y(x_0+\D x, y_0+a \D y)-u_y(x_0,
y_0)\right)
</mrow>
<mrow>
\amp \quad + i\left(v_y(x_0+\D x, y_0+b \D y)-v_y(x_0,
y_0)\right)
</mrow>
</md>.
Because <m>u_y</m> and <m>v_y</m> are continuous at <m>(x_0,y_0)</m>,
<md>
<mrow>
\lim_{ \D z \to 0 } u_y(x_0+\D x, \, y_0+a \, \D y) \amp \ = \
u_y(x_0, \, y_0)
</mrow>
<mrow>
\lim_{ \D z \to 0 } v_y(x_0+\D x, \, y_0+b \, \D y) \amp \ = \
v_y(x_0, \, y_0) \,
</mrow>
</md>,
and so <xref ref="eq_laststepcauchyr"/> goes to 0 as <m>\D z \to 0</m>,
which we set out to prove.
</p>
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