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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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    The key fact here is that the trace map is linear.2012-03-19

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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.