Suppose that f is holomorphic on C and that the real part of f is positive for all z. Show that f is constant?

I assume that you need to show that f is bounded, to use Liouville's Theorem. But I don't see how being bounded below on the real part makes the whole function bounded. I also haven't yet learnt about lattices and/or poles, so should be able to answer the question without using them. Any ideas would be really helpful. Thanks.

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