get that s $^{−}$( s ) and s $^{+}$( s ) are continuous functions of s . From this it follows that ν$_{1}$ and ν$_{2}$ are continuous on [0 , L ]. 

Step 2: From the previous step and the compactness of [0 , L ] we get that ν$_{1}$ ( s ) and ν$_{2}$ ( s ) are both bounded by M$_{ν}$ < π/ 2. We define ϵ$_{ν}$ ≡ π/ 2 − M$_{ν}$ > 0. Fix a t ∈ [0 , L ]. Define ˆ ρ ( s ) by ˆ ρ $^{2}$( s ) = R $^{2}$( s, t ) = || γ ( s ) − γ ( t ) || $^{2}$. Then ˙ ˆ ρ ( s ) = 〈 γ ( s ) − γ ( t ) ˆ ρ , γ $^{′}$( s ) 〉 = 〈 n$_{t}$ ( s ) , γ $^{′}$( s ) 〉 where n$_{t}$ ( s ) = γ ( s ) − γ ( t ) || γ ( s ) − γ ( t ) || = γ ( s ) − γ ( t ) ˆ ρ , the external normal to ∂D ( γ ( t ) , ˆ ρ ) at γ ( s ) (see Figure 4.7). On any interval in s where ˙ ˆ ρ ( s ) > 0 we have that ˆ ρ ( s ) is one to one and strictly increasing. Define s ∗ ≡ s $^{+}$( t ) and s$_{∗}$ ≡ s $^{−}$( t ). We showed above that ˙ ˆ ρ ( s $^{∗}$) = 〈 n$_{t}$ ( s $^{∗}$) , γ $^{′}$( s $^{∗}$) 〉 ≥ cos( M$_{ν}$ ) > 0. 

For 〈 n$_{t}$ ( s ) , γ $^{′}$( s ) 〉 = 0, n$_{t}$ ( s ) and γ ′ will have to have together turned by at least π/ 2 − M$_{ν}$ radians. And until they have turned this far, 〈 n$_{t}$ ( s ) , γ $^{′}$( s ) 〉 > 0. But ˙ n$_{t}$ ( s ) ≤ 1 ρ ≤ 1 r$_{min}$ for some r$_{min}$ > 0. (Choosing r$_{min}$ = r 2 works.) And γ ′ is uniformly continuous on [0 , L ]. Therefore, there is a δ$_{s}$ such that on [ s $^{∗}$, s$^{∗}$+ δ$_{s}$ ], n$_{t}$ ( s ) and γ ′ both turn by less than ϵ$_{ν}$/ 3. Therefore, for s ∈ [ s $^{∗}$, s∗ + δ$_{s}$ ], we have that 〈 n$_{t}$ ( s ) , γ $^{′}$( s ) 〉 > cos( π/ 2 − ϵ$_{ν}$/ 3) and γ ([ s $^{∗}$, s∗ + δ$_{s}$ )) intersects C = ∂D ( γ ( t ) , ρ ) once for each ρ ∈ [ r, r + δ$_{r}$ ], where δ$_{r}$ ≡ δ$_{s}$ cos( π/ 2 − ϵ$_{ν}$/ 3). 

A completely analogous argument works to show that γ ([ s$_{∗}$ − δ$_{s}$, s$_{∗}$ ]) intersects C = ∂D ( γ ( t ) , ρ ) once for each ρ ∈ [ r, r + δ$_{r}$ ]. 

Define d ( t ) to be the distance from D ( γ ( t ) , r ) to γ \ γ ([ s$_{∗}$ − δ$_{s}$, s ∗ + δ$_{s}$ ]). Since γ is TGL, d(t) is greater than zero for all t and is continuous in t . Therefore, there is a smallest distance δ$_{d}$ such that d ( t ) ≥ δ$_{d}$ for all t . Define δ$_{γ}$$_{o}$ = min( δ$_{d}$/ 2 , δ$_{r}$/ 2). 

Therefore, ∂D ( γ ( t ) , ρ ) intersects γ exactly twice for ρ ∈ [ r, r + δ$_{γ}$$_{o}$ ] for any t ∈ [0 , L ]. 

A similar argument shows that ∂D ( γ ( t ) , ρ ) intersects γ exactly twice for ρ ∈ [ r − δ$_{γ}$$_{i}$, r ] for any t ∈ [0 , L ]. Defining δ$_{γ}$ ≡ min( δ$_{γ}$$_{i}$, δ$_{γ}$$_{o}$ ) we get that ∂D ( γ ( t ) , ρ ) intersects γ exactly twice for ρ ∈ [ r − δ$_{γ}$, r + δ$_{γ}$ ], with the additional fact that 〈 n$_{t}$ ( s ) , γ $^{′}$( s ) 〉 > cos( π/ 2 − ϵ$_{ν}$/ 3) 

