diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index 643eb4c..f0739e3 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -34,6 +34,22 @@
               "icon": "lucide-file",
               "title": "3b - Aravind's algorithm and improvements"
             }
+          },
+          {
+            "id": "b51891c564d42ef8",
+            "type": "leaf",
+            "state": {
+              "type": "pdf",
+              "state": {
+                "file": "Concurrent Systems/slides/class 3.pdf",
+                "page": 17,
+                "left": -16,
+                "top": 596,
+                "zoom": 0.9236935866983375
+              },
+              "icon": "lucide-file-text",
+              "title": "class 3"
+            }
           }
         ],
         "currentTab": 1
@@ -210,12 +226,12 @@
   },
   "active": "dd6c003f9d17b0e9",
   "lastOpenFiles": [
-    "Concurrent Systems/notes/3a - Hardware primitives & Lamport Bakery algorithm.md",
+    "Concurrent Systems/slides/class 3.pdf",
     "Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md",
+    "Concurrent Systems/notes/3a - Hardware primitives & Lamport Bakery algorithm.md",
     "Concurrent Systems/notes/1 - CS Basics.md",
     "Concurrent Systems/notes/2b - Round Robin algorithm.md",
     "Concurrent Systems/notes/2 - Fast mutex by Lamport.md",
-    "Concurrent Systems/slides/class 3.pdf",
     "Pasted image 20250310172134.png",
     "Pasted image 20250310103703.png",
     "Concurrent Systems/notes/1 - CS Basics2.md",
diff --git a/Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md b/Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md
index c09f8cf..601d10a 100644
--- a/Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md	
+++ b/Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md	
@@ -91,3 +91,8 @@ lock(i) :=
 	until foreach j != i, STAGE[j] = 0
 ```
 
+-  $p_n$ invokes lock alone, completes its CS and its new DATE is n
+
+**Lemma 1:** DATE's bounds are $[0, n]$
+*Proof:*
+For the upper bound, it's trivial: DATE is either decreased or set to $n$.
\ No newline at end of file