Let $E \subset \mathbb R$, let $M,N \in [0,\infty)$, and let $(f_n)_{n=1}^{\infty}$ and $(g_n)_{n=1}^{\infty}$ be sequences of functions $f_n: E \to \mathbb R$ and $g_n: E \to \mathbb R$. Suppose that for every $n \in \mathbb N$ and every $x \in E$,

\begin{align*} |f_n(x)| \le M. \end{align*}

Suppose also that for every $n \in \mathbb N$ and every $x \in E$,

\begin{align*} |g_n(x)| \le N. \end{align*}

For each $n \in \mathbb N$, define the pointwise sum $f_n+g_n: E \to \mathbb R$ by $(f_n+g_n)(x)=f_n(x)+g_n(x)$. For each $n \in \mathbb N$, define the pointwise product $f_n g_n: E \to \mathbb R$ by $(f_n g_n)(x)=f_n(x)g_n(x)$. For each $a \in \mathbb R$ and each $n \in \mathbb N$, define the scalar multiple $a f_n: E \to \mathbb R$ by $(a f_n)(x)=a f_n(x)$. Then, for every $a \in \mathbb R$, every $n \in \mathbb N$, and every $x \in E$,

\begin{align*} |(f_n+g_n)(x)| \le M+N. \end{align*}

Also,

\begin{align*} |(f_n g_n)(x)| \le MN. \end{align*}

Finally,

\begin{align*} |(a f_n)(x)| \le |a|M. \end{align*}

Consequently, $(f_n+g_n)_{n=1}^{\infty}$, $(f_n g_n)_{n=1}^{\infty}$, and, for each fixed $a \in \mathbb R$, $(a f_n)_{n=1}^{\infty}$ are uniformly bounded on $E$.