I am trying to do a similar proof of Dini's Theorem but with a stricter condition. So, given $f: X \to Y$, where (X, d) and (Y, d') are metric spaces, and
- X is compact
- $f_n \to f$ pointwisely
- $d'(f(x), f_n(x)) \le 2*d'(f(x), f_m(x))$ for all $m, n \in N$ with $n \ge m$
- $f_n$ and f are continuous
So (3) is given instead of the monotonicity of sequence $(f_n)$. I want to show in this case $f_n$ also converges to f uniformly.
Attempt: I am trying to establish an open covering $(A_n^\epsilon$) similar to what is in Dini's Theorem: $A_n^\epsilon=\{d'(f(x),f_n(x))<\epsilon\}$, and then I can have $A_n^\epsilon \subset A_{n+1}^\epsilon$ so that $(A_n^\epsilon)$ forms an open covering of X. Then since X is compact, there is a sub-covering of $(A_n^\epsilon)$ such that all $x \in X$ are in some $A_n^\epsilon$.
But I am struggling to find such appropriate open covering of X. I tried to let $A_n^\epsilon=\{d'(f(x),f_n(x))<\frac{\epsilon}{2^n}\}$ but condition (3) doesn't lead to $A_n^\epsilon \subset A_{n+1}^\epsilon$ as it does in Dini's Theorem.
Can someone please have a look? Any help would be appreciate. Thanks in advance.
Given some $e>0.$ For each $x\in X$
take $m_x \in \Bbb N$ such that $\forall m\ge m_x\,[\,d'(f_m(x),f(x))<e/6\,]$
and take $r_x>0$ such that $d'(f_{m_x}(x), f_{m_x}(z))<e/6$ and $d'(f(x),f(z))<e/6$ for every $z$ in the open ball $B_d(x,r_x).$
Observe that if $n\ge m_x$ and $z\in B_d(x,r_x)$ then $$d'(f_n(z),f(z))\le 2d'(f_{m_x}(z),f(z))\le$$ $$\le 2d'(f_{m_x}(z), f_{m_x}(x))+2d'(f_{m_x}(x),f(x))+2d'(f(x),f(z))<e.$$ Now as $X$ is compact, take a finite $S\subset X$ such that $X=\cup_{x\in S}B_d(x,r_x)$ and let $M=\max\{m_x: x\in S\}.$
Any $z\in X $ belongs to $B_d(x,r_x)$ for some $x\in S$, and $M\ge m_x$ when $x\in S,$ so $$\forall z\in X\, \forall n\ge M\,[\,d'(f_n(z),f(z))<e\,].$$