Exercise of sequence of continuous functions

Question

Let $(f_n)_n$ be a sequence of continuous functions on $D\subset \mathbb{R}^{N} \to \mathbb{R}$ which is monotone decreasing. If $\lim_{n\to\infty }f_n(c))=0$ for some $c\in D$ and $ \epsilon >0$ , show that there are $m \in \mathbb{N}$ and a neighbourhood $U$ of $c$ shuch that if $n>m$ and $x \in U \cap D$ , then $f_n (x)< \epsilon$.

Answer 1

$f_k(c)<\epsilon$ for some $k$. By continuity $f_k(x) <\epsilon$ for all $x$ in some neighborhood. Since $f_n \leq f_k$ for $n \geq k$ we get $f_n(x) <\epsilon$ for all $n \geq k$ for all $x$ in the neighborhood.