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Let E ( · ) denote the expectation of a random variable. We bound from above our probability of interest by a sum of three probabilities as follows. 

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where the last inequality follows from {| $^{∑}$$_{I}$( − ˆ H I UV + E ( ˆ H I $_{UV}$)) | + | $^{∑}$$_{I}$( − E ( ˆ H I $_{UV}$) + H I $_{UV}$) | + | $^{∑}$$_{I}$( − H I UV + H I U + H I V + H$_{UV}$ ) | > | Π | ϵ } ⊆ { ( | $^{∑}$$_{I}$( − ˆ H I UV + E ( ˆ H I $_{UV}$)) | > | Π | ϵ/ 3) ∪ ( | $^{∑}$$_{I}$( − E ( ˆ H I $_{UV}$) + H I $_{UV}$) | > | Π | ϵ/ 3) ∪ | $^{∑}$$_{I}$( − H I UV + H I U + H I V + H$_{UV}$ ) | > | Π | ϵ/ 3 } and Bonferroni’s inequality. 

We will show that the three probabilities (B.6)–(B.8) vanish as N → ∞ ,m → ∞ , m √ N → 0, thus proving the theorem. 

