**Note:** I asked this question before but it wasn’t well written, So I deleted my previous question and re-wrote it.

According to **Dini’s theorem**:

If $X$ is a compact topological space, and ${ f_n }$ is a monotonically

increasing sequence (meaning $f_n(x) leq f_{n+1}(x)$ for all $n$ and $x$) of

continuous real-valued functions on $X$ which converges pointwise to a

continuous function $f$, then the convergence is uniform.The same conclusion holds if ${ f_n }$ is monotonically decreasing

instead of increasing.

(Note: I have proven both cases)

But, what if for every $n$ ${f_n(x0)}$ is monotonic but for some values of $n$ it’s monotonically decreasing and for other it’s monotonically decreasing.

for example; for all even values it is increasing and for non-even values it is decreasing.

**How could I prove that Dini’s theorem is effective in this case?
In other words, how to prove that the convergence is uniform**

Let $A={x: f_n(x) leq f_{n+1}(x) forall n}$ and $B={x: f_n(x) geq f_{n+1}(x) forall n}$. Note that $A$ and $B$ are closed sets and hence they are also compact. Also $A cup B=X$. $f_n to f$ uniformly on each of these sets. Given $epsilon >0$ there exist $n_1, n_2$ such that $|f_n(x)-f(x)| <epsilon$ for all $x in A$ for all $n >n_1$ and $|f_n(x)-f(x)| <epsilon$ for all $x in B$ for all $n >n_1$. Let $n_0=max {n_1,n_2}$. Then $|f_n(x)-f(x)| <epsilon$ for all $x in X$ for all $n >n_0$.