Rényi entropy at infinity or min-entropy
Suppose $p_1 = \max_i p_i$. We have
$$ p_1^n \leq \sum_{i=1}^N p_i^n \leq N p_1^n. $$
Therefore
$$ \frac{n \log p_1 + \log N}{1-n} \leq H_n(X) \leq \frac{n \log p_1}{1-n}. $$
As $n \rightarrow \infty$, $\log N/(1-n) \rightarrow 0$ while $n/(1-n) \rightarrow -1$.