Let $M$ be a Metric compact space. $\{f_n\}_n \subseteq \{f:M \to \mathbb{R}| f$ is continuous$\}$. Suppose that:
- $\lim_{n\to \infty} f_n(x) = 0 \; \forall x \in M$
- $f_1(x)\geq f_2(x)\geq ...\geq f_n(x)\geq ...,\;\forall x \in M$.
Show that $\{f_n\}_n$ is equi-continuous, that is, $\forall \epsilon>0$, $\forall x \in M$ $\exists\; \delta(\epsilon,x)$ s.t. if $|x-y|<\delta(\epsilon,x) \implies|f_k(x)-f_k(y)|\leq \epsilon \;\forall k$.
I believe I've shown this result without the necessity that $f_i(x)\geq f_{i+1}(x)\;\forall x$ which is concerning because it's unlikely that my professor made this restriction for nothing. So I need to find my mistake.
Lets fix a value of $x \in M$. Let $\epsilon>0$.
As $\lim_kf_k(x) = 0$, exists a $k_0$ such that $|f_k(x)|\leq \frac{\epsilon}{2} \; \forall k \geq k_0$. So $|f_k(x)-f_k(y)|\leq |f_k(x)|+|f_k(y)|\leq \epsilon \forall x,y \in M$ if $k \geq k_0$.
Let us look at $k < k_0$ then.
Every $f_n$ is continuous, therefore $\exists \delta_n(\epsilon,x)$ such that if $|x-y|\leq \delta_n(\epsilon,x) \implies |f_n(x)-f_n(y)|\leq \epsilon$ for $n\in \{1,2,3,...,k_0-1\}$. Take $\delta = min_n\{\delta_n(\epsilon,x)\}$.
Then if $|x-y|\leq \delta \implies |f_k(x)-f_k(y)|\leq \epsilon\; \forall k$
So $\{f_n\}_n$ is equi-continuous.
Where is my mistake ????