Quartz sync: May 26, 2025, 4:41 PM

content/Revision/Real Analysis/Question 10.md

@@ -0,0 +1,17 @@
+# Question
+What is a net? Given an example of an upward filtered ordered set.
+# Answer
+A net is a function with a domain that has a directed set.
+
+A net in $X$ is denoted by $x_{\bullet} = (x_{i})_{i \in I}$
+And is a function of the form $x_{\bullet} : I \to X$
+
+> [!info] What is the domain in a function?
+> The function $f : X \to Y$ would have the domain $X$. ($Y$ would be the codomain.)
+
+## Directed Set
+A directed set is a non-empty set $I$ where the preorder (can be though as a direction) is usually assumed to be $\leq$,which is upward directed.
+
+So if you have $a,b \in A$ there exists some $c \in I$ such that
+- $a \leq c$
+- $b \leq c$

content/Revision/Real Analysis/Question 11.md

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+# Question

content/Revision/Real Analysis/Question 7.md

@@ -16,7 +16,7 @@ If an [[Open Cover|open cover]] of $A$ has no finite [[Subcover|subcover]], then
 
 The centres of these cubes form a [[Cauchy Sequence|Cauchy sequence]] with a limit $x \in A$.
 
-![[Pasted image 20250130121836.png]]
+![[Pasted image 20250130120238.png]]
 $$d(x_{n}, x_{m}) \lt \varepsilon, \ \forall n,m \gt N$$
 Now we have to show that $R^{n}$ is complete.
 

content/Revision/Real Analysis/Question 9.md

@@ -1,3 +1,11 @@
 # Question
 Why does a real valued continuous function obtain its maximum on a compact set?
-# Answer
+# Answer
+Let's say we have
+- a [[Continuous|continuous]] function $f : X \to \mathbb{R}$
+- a [[Compact|compact]] set $X$
+## Proof/Reasoning
+- The [[Continuous|continuous]] image of a [[Compact|compact set]] is [[Compact|compact]]
+	- Because inverse images of an [[Open Cover|open cover]] will again be an open cover.
+- Then as a consequence of the [[Heine-Borel]] theorem will allow us to get a maximum on a compact set.
+	- Because a continuous image of a compact set