Show a sequence of functions is pointwise but not uniformly convergent 
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a bounded function with the properties $f(0)=0$ $\lim_{x\to\infty} f(x)=1$. For each $n\in\mathbb{N}$ define $f_n(x)=f(x+e^n)$ with $(x\in\mathbb{R})$
Show that $(f_n)_{n\in\mathbb{N}}$ is pointwise but not uniformly convergent

So to prove pointwise, I would do the following:
$f_n(x)=f(x+e^n)$ so as $n\rightarrow\infty$ you would have $e^n \rightarrow\infty$ So I am unsure how to prove this? What would the limit function be?

For each $x \in \Bbb R$, $\lim_{n \to \infty} x + e^n = \infty$,
and  $\lim_{t\to\infty} f(t)=1$, therefore:
 $$
\lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} f(x + e^n) = 1 \, .$$
 So $(f_n)$
converges pointwise to the function $F(x) = 1$.
To prove that the convergence is not uniform, compute
$f_n(x) - F(x)$ for $x = -e^n$.
