limit of sequences of functions and uniform convergence I have two sequences of functions:
$$ h_n(x) = (x-\frac1n)^2 \qquad \text{for   } x \in [0,1]
$$ 
$$k_n(x) = (x-\frac1n)^2 \qquad \text{for   } x \in \mathbb R
$$
I need to find their limits and determine whether the convergence is uniform.
How are these two sequences different?

$h_n$ are on a compact sets, while $k_n$ are on the whole real line.
For uniform convergence, it can make a difference: the sequence $f_n(x)=x/n$ ($x\in \mathbb{R}$) converges to $f\equiv 0$ on any compact, but not uniformly since
$$\sup_{x\in \mathbb{R}} |f_n(x)|\equiv +\infty \ for\ any\ n,$$
while for any compact $K$
$$ \sup_{x\in K}|f_n(x)|\leq \sup_{y\in K} |y|/n\to 0.  $$
You situation is quite similar.
