Index bound for subsequences

Lemma. Let $(n_k)$ be a strictly increasing sequence of positive integers, i.e., $n_1<n_2<\cdots$. Then

Proof. By induction. For $k=1$, $n_1\ge 1$. Assume $n_k\ge k$. Then $n_{k+1}>n_k\ge k$, hence $n_{k+1}\ge k+1$.

This estimate is often used when transferring “eventually” statements from a sequence to a subsequence (e.g., to compare thresholds $N$ and $K$).