Give an example of a sequence of continuous functions $f_n:[0,1]\rightarrow [0,1]$ such that $\lim_{n\rightarrow\infty}m(E_n(\varepsilon)) = 0$ for every $\varepsilon >0$ but $\lim_{n\rightarrow\infty}f_n(x)=0$ for no $x$.
Where $E_n(\varepsilon) := \{x\in [0,1] : |f_n(x)|\geq \varepsilon\}$
My intuition for this is that I need this sequence of functions $\{f_n\}$ to map larger and larger sections of $[0,1]$ closer and closer to zero as $n$ increases. While at the same time never allowing any $x$ to actually drop down to zero in the limit. This seems impossible to me! I mean if most of the points in $[0,1]$ must get arbitrarily close to zero, for large enough $n$, this is the very definition of going to zero in the limit.
Ok so even though it seems impossible, I've still be trying to find a function using my bag of tricks:
- raising functions to $\frac{1}{n}$
- speeding up the oscillating velocity of sine
- looking at sections of the exponential function farther and farther down the negative real axis
- making functions out of the harmonic series or other infinite series
- trying to show such a function must exist rather than actually constructing one
Anyways, no luck so far. Maybe just a hint would be good for now and then I could come back if I need more help. Thanks.