diff --git a/content/Math4202/Math4202_L23.md b/content/Math4202/Math4202_L23.md index 8733d46..0582521 100644 --- a/content/Math4202/Math4202_L23.md +++ b/content/Math4202/Math4202_L23.md @@ -61,8 +61,7 @@ Therefore $f$ must have a root in $B^2$. If $\|a_{n-1}\|+\|a_{n-2}\|+\cdots+\|a_0\|< R$ has a root in the disk $B^2_R$. (and $R\geq 1$, otherwise follows part 1) -Consider $\tilde{f}(x)=f(Rx)$. - +Consider $\tilde{f}(x)=f(Rx)$. $$ \begin{aligned} \tilde{f}(x) @@ -73,7 +72,7 @@ $$ $$ \begin{aligned} -\|\frac{a_{n-1}}{R}\|+\|\frac{a_{n-2}}{R^2}\|+\cdots+\|\frac{a_0}{R^n}\|&=\frac{1}{R}\|a_{n-1}\|+\frac{1}{R^2}\|a_{n-2}\|+\cdots+\frac{1}{R^n}\|a_0\|\\ +\left\|\frac{a_{n-1}}{R}\right\|+\left\|\frac{a_{n-2}}{R^2}\right\|+\cdots+\left\|\frac{a_0}{R^n}\right\|&=\frac{1}{R}\|a_{n-1}\|+\frac{1}{R^2}\|a_{n-2}\|+\cdots+\frac{1}{R^n}\|a_0\|\\ &<\frac{1}{R}\left(\|a_{n-1}\|+\|a_{n-2}\|+\cdots+\|a_0\|\right)\\ &<\frac{1}{R}<1 \end{aligned}