updates

content/Math4111/Math4111_L5.md

@@ -30,7 +30,8 @@ $\forall x\in \mathbb{R}_{>0},\forall n\in \mathbb{N},\exist$ unique $y\in \math
 
 (Because of this Theorem we can define $x^{1/x}=y$ and $\sqrt{x}=y$)
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 We cna assume $n\geq 2$ (For $n=1,y=x$)
 
@@ -94,7 +95,7 @@ So want $k\leq \frac{y^n-x}{ny^{n-1}}$
 
 [For actual proof, see the text.]
 
-QED
+</details>
 
 ### Complex numbers
 
@@ -151,7 +152,8 @@ $$
 (\sum a_j b_j)^2=(\sum a_j^2)(\sum b_j^2)
 $$
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 For real numbers:
 
@@ -169,7 +171,7 @@ let $t=C/B$ to get $0\leq A-2(C/B)C+(C/B)^2B=A-\frac{C^2}{B}$
 
 to generalize this to $\mathbb{C}$, $A=\sum |a_j|^2,B=\sum |b_j|^2,C=\sum |a_j \bar{b_j}|$.
 
-QED
+</details>
 
 ### Euclidean spaces