Proof that a function sequence does not converge uniformly

Question

Suppose we have a sequence of functions defined as :

$f_n(t) = \cases{0 & for $t \leq 0$ \\ t^n & for $0 < t < 1$\\ 1 & for $t \geq1$ \\ }$

Pointwise, it converges to

$f(t) = \cases{0 & for $t < 1$\\ 1 & for $t \geq 1$ \\ }$

But how can you show (formally) that it does not converge uniformly?

Answer 1

Suppose the convergence is uniform. Choose $\epsilon = \frac 12$. There must exist $N$ so that $n \ge N$ implies $|f_n(t) - f(t)| < \frac 12$ for all $t \in [0,1]$. In particular you must have $t^N < \frac 12$ for all $t \in [0,1)$. This is false for any $N \in \mathbf N$.

Answer 2

The uniform limit of a sequence of continuous functions is again continuous, but the $\{f_n\}$ are continuous while $f$ is not.