deletions | additions       diff --git a/Making_a_Spanner_Proof_by__.md b/Making_a_Spanner_Proof_by__.md  index e19c05a..64170cd 100644  --- a/Making_a_Spanner_Proof_by__.md  +++ b/Making_a_Spanner_Proof_by__.md 
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We now move onto the pairs of points containing `k`. Let us refer to one such arbitrary pair as `i`,`k`. We refer to the path as \(i,v_1\dots,j,k\).  We have that \(p(i,k) = p(i,j) + |jk|\). Clearly \(j\neq k\), hence by our inductive hypothesis \(p(i,j)\leq t\cdot |ij|\). |ij|\) (1).  Furthermore we claim that \(|ij| < |ik|\). This holds by the way we construct our graph. `k` is clearly in the same segment as `k`. Therefore,.... (2).  As such, from (1) we have