content_image/304768.jpg content_image/304769.jpg content_image/304770.jpg content_image/304771.jpg
Let z ∈ ℓ $^{∞}$( F ) be ‖·‖ -uniformly continuous. If Π is totally bounded, then h is continuous at ( z, 0) . 

Proof of Lemma A.5 and Corollary A.6. Fix a sequence { ( z$_{m}$, r$_{m}$ ) }$_{m}$$_{≥}$$_{1}$ satisfying the condi- tions of the Lemma A.5. Observe that, for every m ≥ 1, 

<img src='content_image/304768.jpg'>

where the above RHS expression is o (1) because z$_{m}$ → z in ℓ $^{∞}$( F ). Therefore, 

<img src='content_image/304769.jpg'>

Let us show now that the RHS expression in (A.4) is o (1). 

For any r > 0, there exists a π$_{r}$ ∈ Π$_{r}$ such that 

Furthermore, there exists a π ⋆ r ∈ Π ⋆ such that 

<img src='content_image/304770.jpg'>

<img src='content_image/304771.jpg'>

