diff --git a/06-hash/hash.tex b/06-hash/hash.tex index e9c4fe4a7841239f8c9248633f423623dede04a6..8d6b704a27c914832885c7c264db139b36e220be 100644 --- a/06-hash/hash.tex +++ b/06-hash/hash.tex @@ -806,9 +806,9 @@ $$ We bound the former probability simply by~1 and use the previous corrolary to bound the latter probability: $$ -\E[\vert R\vert] \le 3 + \sum_{\ell\ge 0} 2^{\ell+3} \cdot q^{2^\ell} - = 3 + 8 \cdot \sum_{\ell\ge 0} 2^\ell \cdot q^{2^\ell} - \le 3 + 8 \cdot \sum_{i\ge 1} i\cdot q^i. +\E[\vert R\vert] \le 3 + \sum_{\ell\ge 0} 2^{\ell+3} \cdot 12 \cdot q^{2^\ell} + = 3 + 8 \cdot 12 \cdot \sum_{\ell\ge 0} 2^\ell \cdot q^{2^\ell} + \le 3 + 96 \cdot \sum_{i\ge 1} i\cdot q^i. $$ Since the last sum converges for an arbitrary $q\in(0,1)$, the expectation of~$\vert R\vert$ is at most a~constant. This concludes the proof of the theorem.