Theorem. If the sequence $f_1,\,f_2,\,f_3,\,\ldots$ of real functions converges at each point of the interval $[a,\,b]$ but does not converge uniformly on this interval, then there exists at least one point $x_0$ of the interval such that the function sequence converges uniformly on no closed sub-interval of $[a,\,b]$ containing $x_0$ .

Proof. Let the limit function of the sequence on the interval $[a,\,b]$ be $f$ . According the entry uniform convergence on union interval, the sequence can not converge uniformly to $f$ both half-intervals $[a,\,\frac{a+b}{2}]$ and $[\frac{a+b}{2},\,b]$ , since otherwise it would do it on the union $[a,\,b]$ . Denote by $[a_1,\,b_1]$ the first (fom left) of those half-intervals on which the convergence is not uniform. We have $[a,\,b] \supset [a_1,\,b_1]$ . Then the interval $[a_1,\,b_1]$ is halved and chosen its half-interval $[a_2,\,b_2]$ on which the convergence is not uniform. We can continue similarly arbitrarily far and obtain a unique endless sequence $$[a,\,b] \supset [a_1,\,b_1] \supset [a_2,\,b_2] \supset \ldots$$ of nested intervals on which the convergence of the function sequence is not uniform, and besides the length of the intervals tend to zero: $$\lim_{n\to\infty}(b_n-a_n) = \lim_{n\to\infty}\frac{b-a}{2^n} = 0.$$ The nested interval theorem thus gives a unique real number $x_0$ belonging to each of the intervals $[a,\,b]$ and $[a_n,\,b_n]$ . Then $\lim_{n\to\infty}a_n = x_0 = \lim_{n\to\infty}b_n$ . Let us choose $\alpha$ and $\beta$ such that $a \leqq \alpha \leqq x_0 \leqq \beta \leqq b$ . There exist the integers $n_1$ and $n_2$ such that $$|a_n-x_0| = x_0-a_n \leqq x_0-\alpha\quad \mbox{when}\;\;n > n_1$$ $$|b_n-x_0| = b_n-x_0 \leqq \beta-x_0\quad \mbox{when}\;\;n > n_2.$$ Therefore $$\alpha \leqq a_n \leqq x_0 \leqq b_n \leqq \beta \quad\mbox{when}\;\;n > \max\{n_1,\,n_2\}.$$ This means that $f_n \to f$ not uniformly on $[a_n,\,b_n] \subset [\alpha,\,\beta]$ , whence the function sequence does not converge uniformly on the arbitrarily chosen subinterval $[\alpha,\,\beta]$ of $[a,\,b]$ containing $x_0$ .