Confusion regarding uniform boundedness

Question

I am estimating a sequence of functions and struggling with the following estimations.

Suppose $F_n:[0,1]\rightarrow (0,\infty)$, a continuous sequence of functions. We know the following

For an arbitrary $n\in {\mathbb{N}}$ and $x_0\in [0,1]$, there exists an $\epsilon_n>0$ (depending only on n not on $x_0$), and two positive constants $C_{1,x_0},C_{2,x_0}$(only depending on $x_0$, not on n) such that $$ C_{1,x_0}<F_n(x)<C_{2,x_0}\,\forall x\in B(x_0,\epsilon_n):=(x_0-\epsilon_n,x_0+\epsilon_n). $$ Is $F_n$ uniformly bounded on $[0,1]$ by two positive constants?

My approach: I was trying to use a compactness argument for this. I first fixed a $n$ then, on each $B(x_0,\epsilon_n)$, $F_n$ is bounded by positive constants, not depending on $n$. So, after taking finite subcover, I tried to establish the claim. However, the issue is if I take the finite subcover then the points $x_0$ will start to depend on $n$.

Is there any solution for this? Or is the claim false? Any suggestion would be of great help.

Answer 1

I think your claim is wrong. As you mentioned, once taking the finite subcover to determine the uniform constants, it starts depending on $n$. The counter-example lies there - We should take a series of functions, for which there is a "bad point" (i.e., one which requires a high constant), but this "bad point" is different each time. A concrete example would be -

$$ F_n(x) = n^2x*1_{x\le \frac{1}{n}} +(n-n^2(x-\frac{1}{n}))*1_{\frac{1}{n}\le x\le\frac{2}{n}} $$

Geometrically, the function increases linearly from $(0,0)$ to $(\frac{1}{n},n)$, and then decreases linearly to $(\frac{2}{n}, 0)$.

For every $x_0$, there are such constants $C_{1,x_0}, C_{2,x_0}$, which bound all functions around $x_0$ (but for each function, in a different neighborhood. which in the case of $x_0=0$ is shrinking rapidly with $n$), but the series is of course not uniformly bounded (as $F_n(\frac{1}{n})\rightarrow\infty)$.

Edit:

Now I see that this is a duplicate - A sequence of functions $\{f_n(x)\}_{n=1}^{\infty} \subseteq C[0,1]$ that is pointwise bounded but not uniformly bounded. consider closing.