diff --git a/main.pdf b/main.pdf
index f2674ec..05359de 100644
Binary files a/main.pdf and b/main.pdf differ
diff --git a/main.tex b/main.tex
index 47f9dda..194ecf2 100644
--- a/main.tex
+++ b/main.tex
@@ -160,7 +160,7 @@ s.t.&   &   (x_i-x_j)^2 + (x_j-x_k)^2&\geq (x_i-x_k)^2  &   &\forall i,j,k\in V\
 \end{aligned}
 \end{equation*}
 
-This SDP models \uscut{} since every assignment of $x$ corresponds to a cut and the objective is the sparsity of the cut (up to a constant factor, but we don't care since we cannot achieve a constant factor approximation anyway). Now we consider a relaxation which is similar to \lp{}.
+This SDP models \uscut{} since every assignment of $x$ corresponds to a cut and the objective is the sparsity of the cut (up to a constant factor, but we don't care since we cannot achieve a constant factor approximation anyway). Consider a relaxation which is similar to \lp{}.
 
 \begin{equation*}
 \begin{aligned}
@@ -171,12 +171,12 @@ s.t.&   &                   \sum_{ij\in V\times V}\|v_i-v_j\|^2&=1  &   &\\
 \end{aligned}
 \end{equation*}
 
-To get a $O(\sqrt{\log n})$ (randomized) approximation algorithm we need to first solve the SDP and then round the solution to get a cut $\delta(S)$ with $c(\delta(S))=|S| \opt(SDP) O(n\sqrt{\log n})$. If we can find two sets $S,T\subset V$ both of size $\Omega(n)$ that are well-separated, in the sense that for any $s\in S$ and $t\in T$, $\|v_s-v_t\|^2=\Omega(1/\sqrt{\log n})$, then we have
+To get a $O(\sqrt{\log n})$ (randomized) approximation algorithm we need to first solve the SDP and then round the solution to get a cut $\delta(S)$ with $c(\delta(S))=|S| \opt(SDP) O(n\sqrt{\log n})$. If there are two sets $S,T\subset V$ both of size $\Omega(n)$ that are well-separated, in the sense that for any $s\in S$ and $t\in T$, $\|v_s-v_t\|^2=\Omega(1/\sqrt{\log n})$, then the SDP gap follows from
 
 \[
 \frac{c(\delta(S))}{|S||V-S|}
-\leq n|S| \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{\sum_{i\in S,j\in T} \|v_i-v_j\|^2}
-\leq |S| \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{n} O(\sqrt{\log n}) 
+\leq \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{\sum_{i\in S,j\in T} \|v_i-v_j\|^2}
+\leq \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{n^2} O(\sqrt{\log n}) 
 \leq O(\sqrt{\log n}) \opt(SDP).
 \]