Why is the trace map from a finite extension of $\mathbb{Q}_p$ continuous?

Let $k$ be a finite extension of $\mathbb{Q}_{p}$. Why is $tr_{k/\mathbb{Q}_{p}}$ a continuous map from $k$ onto $\mathbb{Q}_{p}$?

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Have you tried something? – Patrick Da Silva Mar 19 '12 at 3:00
You need to put together a few facts, but this shouldn't be so bad. What's your favorite definition of the trace? – Dylan Moreland Mar 19 '12 at 3:03
The key fact here is that the trace map is linear. – Álvaro Lozano-Robledo Mar 19 '12 at 3:16

Show that a $\mathbb Q_p$-linear map $f:V\to W$ of finite dimensional $\mathbb Q_p$-vector spaces is always continuous. What you want then follows from $\mathbb Q_p$-linearity of the trace, which is more or less immediate.