2.7.13 changed an equality to a inequality

chapters/chapter2/chapter2-7.tex

@@ -230,8 +230,8 @@ \section{Properties of Infinite Series}
 \begin{align*}
   \abs{\sum_{j={m+1}}^n x_jy_j} 
   &= \abs{s_ny_{n+1} - s_my_{m+1} + \sum_{j=m+1}^n s_j(y_j - y_{j+1})}\\
-  &= \abs{s_ny_{n+1} - s_my_{m+1}} + \abs{\sum_{j=m+1}^n s_j(y_j - y_{j+1})} \tag{$\bigtriangleup$ inequality}\\
-  &= \abs{(s_n - s_m)y_{m+1}} + \abs{\sum_{j=m+1}^n s_j(y_j - y_{j+1})} \tag{$y_{m+1} > y_{n+1}$}\\
+  &\leq \abs{s_ny_{n+1} - s_my_{m+1}} + \abs{\sum_{j=m+1}^n s_j(y_j - y_{j+1})} \tag{$\bigtriangleup$ inequality}\\
+  &\leq \abs{(s_n - s_m)y_{m+1}} + \abs{\sum_{j=m+1}^n s_j(y_j - y_{j+1})} \tag{$y_{m+1} > y_{n+1}$}\\
   &\leq M\abs{y_{m+1}} + M \abs{y_{m+1} - y_{n+1}}\\
   &\leq 2M\abs{y_{m+1}}
 \end{align*}
@@ -290,4 +290,4 @@ \section{Properties of Infinite Series}
 then $A < \epsilon/(2b_1)$ and $\abs{\sum_{j={m+1}}^n x_jy_j} < 2\abs{b_1}\cdot \frac{\epsilon}{2\abs{b_1}}$,
 which means we have the Cauchy Criterion for $\sum x_ny_n$, and therefore it converges.
 }
-}
+}