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