updates

content/Math4202/Math4202_L23.md

@@ -29,6 +29,8 @@ $f|_{B^2}$ is a continuous map from $B^2\to \mathbb{R}^2-\{0\}$.
 
 $f|_{S^1=\partial B^2}:S^1\to \mathbb{R}-\{0\}$ **is nulhomotopic**.
 
+> Recall that: Any map $g:S^1\to Y$ is nulhomotopic whenever it extends to a continuous map $G:B^2\to Y$.
+
 Construct a homotopy between $f|_{S^1}$ and $g$
 
 $$
@@ -57,7 +59,7 @@ Therefore $f$ must have a root in $B^2$.
 <details>
 <summary>Proof: part 2</summary>
 
-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)
+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 $\tilde{f}(x)=f(Rx)$.