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I have to prove finite blaschke products are the examples of proper holomorphic maps of the unit disc into itself.

I know formal definition of proper map so I took any compact set on $\mathbb{D}$ and try to prove it's preimage is compact. I have no idea how to do it.

I think it may be very trivial but no idea. Any help?

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    The pre image is closed in $\mathbb D$ by continuity. Try to show it must be contained in some $\overline {D(0,r)}, r<1.$ – 2017-01-25
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    @zhw. I think it is not obvious. Could u do for me? – 2017-01-25
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    So is the exercise to show that $f$ is proper iff $f$ is a finite Blaschke product? My comment only addressed $\impliedby$ – 2017-01-25
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    yes you are correct. I know one way I only asking another way – 2017-01-26

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