Uniform convergence sequence of positive reals

Question

Suppose $f_n$ for $n ∈ \Bbb N$ and $f$ are functions on $J$. If there exists a sequence $(a_n)$ of positive reals satisfying $a_n→0$ as $n→∞$ and $$|f_n(x)−f(x)|≤a_n, \quad ∀n∈\Bbb N, ∀x∈J$$ then $(f_n)$ converges uniformly to $f$.

Can someone help me with the proof of this theorem?

Answer 1

$$(\forall n\in \Bbb N)\;(\forall x\in J)\;$$ $$|f_n(x)-f(x)|\le a_n \implies $$

$$(\forall n\in \Bbb N)$$ $$ 0\le \sup_{x\in J}|f_n(x)-f(x)|\le a_n$$

and squeeze.

Answer 2

Let $\epsilon>0.$ Let $N$ be large enough so that $n>N$ implies $a_n<\epsilon.$ Then for $n>N$, $$ |f_n(x)-f(x)|\leq a_n < \epsilon. $$