for reaching out about this exercise. It looks like you are trying to prove the uniform convergence of a sequence of functions. To do this, you will need to show that for any given $\epsilon>0$, there exists an $N \in \mathbb{N}$ such that $|f_n(t)|<\epsilon$ for all $n \geq N$ and for all $t \in [0,1]$.
One approach to proving this is to use the Weierstrass M-test. This theorem states that if there exists a sequence of positive numbers $M_n$ such that $|f_n(t)| \leq M_n$ for all $n$ and for all $t \in [0,1]$, and if $\sum_{n=1}^{\infty} M_n$ converges, then the series $\sum_{n=1}^{\infty} f_n(t)$ converges uniformly on $[0,1]$.
In this case, we can choose $M_n = 1$ for all $n$, since $|f_n(t)| \leq 1$ for all $t \in [0,1]$. And since $\sum_{n=1}^{\infty} 1$ is a convergent series, by the Weierstrass M-test, we can conclude that the sequence $f_n$ converges uniformly to the null function on $[0,1]$.
I hope this helps with your exercise. Let me know if you have any other questions or if you need further clarification on any of the steps. Good luck!
