point preventing uniform convergence

Theorem. If the sequence  $f_1,f_2,f_3,\mathrm{\dots }$  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

$$[a,b]\supset [a_1,b_1]\supset [a_2,b_2]\supset \mathrm{\dots }$$

of nested intervals on which the convergence of the function sequence is not uniform, and besides the length of the intervals tend to zero:

$$\underset{n\to \mathrm{\infty }}{lim}(b_n-a_n)=\underset{n\to \mathrm{\infty }}{lim}\frac{b-a}{2^n}=0.$$

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  ${lim}_{n\to \mathrm{\infty }}{a}_n=x_0={lim}_{n\to \mathrm{\infty }}{b}_n$.  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

$$|a_n-x_0|=x_0-a_n\leqq x_0-\alpha \mathit{\hspace{1em}}\text{when}n>n_1$$

$$|b_n-x_0|=b_n-x_0\leqq \beta -x_0\mathit{\hspace{1em}}\text{when}n>n_2.$$

Therefore

$$\alpha \leqq a_n\leqq x_0\leqq b_n\leqq \beta \mathit{\hspace{1em}}\text{when}n>\max \{n_1,n_2\}.$$

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 $x_0$.