Why is this sequence of functions not uniformly convergent?
Solution 1:
You can't just take $n=\infty$. In effect what you have done is to verify pointwise convergence, not uniform convergence.
Uniform convergence fails because, for every $n$, $\sup_{x\in[0,1]} |f_n(x)-f(x)| = 1$, which you can see by continuity.
Alternatively, uniform convergence must fail because each $f_n$ is continuous, $f$ is not continuous, and a uniform limit of continuous functions is continuous.