Why this space is not a complete space with this norm

Question

Show that the space $C_0(\mathbb{R})$ of all the real continuous functions $f:\mathbb{R} \to \mathbb{R}$ with compact support is not a complete space with the norm $||f||= \sup_{t∈ \mathbb{R}}|f(t)|$.

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Answer 1

Let $f(x) = \frac 1{1+x^2}$, define $f_n\colon \mathbb R \to \mathbb R$ by $f_n(x) = f(x)$ for $x \in [-n,n]$, $f_n(x) = 0$ for $x \not\in [-n-1, n+1]$, affine-linear in between. Show that $(f_n)$ is Cauchy, but not convergent to some compact support function (as its limit in $\ell^\infty(\mathbb R)$ is $f$).