diff --git a/nearby_lagrangian_counterexample_v2.txt b/nearby_lagrangian_counterexample_v2.txt
index eaa8306..1115035 100644
--- a/nearby_lagrangian_counterexample_v2.txt
+++ b/nearby_lagrangian_counterexample_v2.txt
@@ -177,7 +177,7 @@ SECTION 2. THE MAIN THEOREM
         is even and at least some explicit bound (below, N = 2m with
         m >= 20 and with a further open condition m leaves room of at least
         ten dimensions; the construction is stable in m). No reader should
-        worry about the exact value of N; every inequality we use has
+        worry about the exact value of N; every ineuality we use has
         plenty of slack.
 
     The proof will proceed in the order of the plan of Section 1.4. We
@@ -522,7 +522,7 @@ SECTION 3. STABLE TUBE DATA AND A NONZERO CLASS
     QED.
 
     Why this class really is nonzero. The only fact that matters
-    numerically is the strict inequality
+    numerically is the strict ineuality
 
         |image(partial)| <= 4 < 32 = |pi_9(H^s(*))_(2)|.
 
@@ -594,7 +594,7 @@ SECTION 3. STABLE TUBE DATA AND A NONZERO CLASS
     is positive on the compact set B x S^(2m-1). It therefore has a
     positive minimum c and a finite maximum C. Using the polar gradient
     formula grad g_b(r u) = r (2 h_b(u) u + grad_S h_b(u)) from Section
-    3.1, we get the first inequality c|w| <= |grad_w g_b(w)| <= C|w| for
+    3.1, we get the first ineuality c|w| <= |grad_w g_b(w)| <= C|w| for
     all w = r u. The extension at w = 0 is continuous with value zero
     because both sides vanish at w = 0.
 
@@ -736,7 +736,7 @@ SECTION 4. A FAMILY WITH ONE CRITICAL POINT HAS ZERO TUBE CLASS
         df_b(d/dr) = 2 f_b / r < 0   (since f_b < 0 on the negative side there).
 
     The conditions "gradient points so that f_b decreases" and "radial
-    outward near r = R_0" are strict linear inequalities in the field;
+    outward near r = R_0" are strict linear ineualities in the field;
     convex combinations preserve them, so we can interpolate smoothly and
     also uniformly over b. The flow of this field cannot exit through
     { f_b = -delta } (it strictly decreases f_b) and must reach the sphere
@@ -1102,8 +1102,8 @@ SECTION 5. THE ZERO-LEVEL COBORDISM ARGUMENT
     need ds_b(V) = +-epsilon |grad h|^2 > 0, which holds because dh is
     nonzero near h = 0 (zero is a regular value). Away from the prescribed
     regions, use strict descent/ascent for f_b. All the conditions are strict
-    linear inequalities in the field; convex patching (partition of unity)
-    preserves every applicable inequality and keeps the displayed
+    linear ineualities in the field; convex patching (partition of unity)
+    preserves every applicable ineuality and keeps the displayed
     prescription on smaller end collars. Everything is smooth in the compact
     parameter space. QED.