If a real-valued sequence of functions is uniformly bounded , then it is pointwise bounded.

Question

Show that if $\{f_n\}$ is a uniformly bounded real-valued sequence, then it is pointwise bounded.

It seems quite obvious that $\mid f_n (x)\mid \leq M, \forall n,x$ means that for every $x, \mid f_n(x)\mid \leq M, \forall n$. How can I formally prove it?

Answer 1

Let $D$ be the domain of your functions. You are assuming that there is a number $M>0$ such that$$(\forall x\in D)(\forall n\in\mathbb N):\bigl\lvert f_n(x)\bigr\rvert\leqslant M.\tag1$$Now, take $x\in D$. It follows from $(1)$ that$$(\forall n\in\mathbb N):\bigl\lvert f_n(x)\bigr\rvert\leqslant M.$$