Let $f_n : \mathbb{R} \to\mathbb{R}$ be continuous functions for n ≥ 1. Suppose foreach $x \in \mathbb{R}$, there exists $n(x) \in\mathbb{ N}$ such that $f_{n(x)}(x) = x + n(x).$Prove that there exists $a < b$ and $m \in \mathbb{N}$ such that $f_m(x) = x+m$for all $x \in [a, b].$
my idea
suppose for all $[a,b]$ type intervals above is not rue then take $[a_1,b_1]$we can find $x'_1<x^{''}_1$ such that $x'_1,x^{''}_1\in [a_1,b_1]$ with $x'_1\neq x^{''}_1$ and $n(x'_2)\neq n(x^{''}_2)$.
choose another subset $[a_2,b_2]\subset [a_1,b_1]$ such that $x'_1\in [a_2,b_2]$ and $x^{''}_1\notin [a_2,b_2]$and points $x'_2$ with $x'_2\in [a_2,b_2]$ such that $n(x'_2)\neq n(x^{'}_1)$.
this will give us a sequence ${x'_n}$ which has a convergent subsequence ${x'_{n_k}}$ which converges to $l$ hence$\lim_{k\to\infty} f_{n(x'_{n_k})}(x'_{n_k}) =\lim_{k\to\infty} (x'_{n_k} + n(x'_{n_k}))=l+n(l)$,but still unable to move forward