real analysis - Uniform convergence of sequence of functions with infinite roots to a limit with finite roots

Consider a sequence of continuous functions $(f_n)$ defined over $[0,1]$ such that, for all $n$, the set:

$$A_n = \{x\in [0,1] : f_n(x) = 0\}$$

is infinite in cardinality. Can $(f_n)$ uniformly converge to some continuous limit $f$ which only has finitely many zeroes? What if $f$ only has finitely many crossings, and no other zeroes, and for each $f_n$ there are infinitely many crossings (and no other zeroes)?

Answer

For the first question. For $n>2$ let
$$

f_n(x)=\begin{cases}\dfrac{1/2-1/n-x}{1/2-1/n} &0\le x\le1/2-1/n\\
0 & 1/2-1/n\dfrac{x-1/2-1/n}{1/2-1/n} & 1/2+1/n\le x\le1\end{cases}
$$
Then $\{x\in[0,1]:f_n(x)=0\}=[1/2-1/n,1/2+1/n]$ and $f_n$ converges uniformly to $|2\,x-1|$.

For the second, you can use the same type of function changing its value on the interval $[1/2-1/n,1/2+1/n]$ to
$$\frac1n\,(1/2-1/n-x)(x-1/2-1/n)\sin\frac{1}{(1/2-1/n-x)(x-1/2-1/n)}.$$