Let $E \subset \mathbb{R}$, and let $(f_n)_{n=1}^{\infty}$ be a sequence of functions $f_n: E \to \mathbb{R}$. Suppose that $(f_n)_{n=1}^{\infty}$ is uniformly bounded on $E$; that is, there exists a constant $M \geq 0$ such that
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\begin{align*}
|f_n(x)| \leq M
\end{align*}
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for every $n \in \mathbb{N}$ and every $x \in E$. Then for every strictly increasing sequence $(n_k)_{k=1}^{\infty}$ in $\mathbb{N}$, the subsequence $(f_{n_k})_{k=1}^{\infty}$ is uniformly bounded on $E$ with the same common bound $M$.