sum of real and imaginary parts constant?

Hey, my flat-mates left this problem on the fridge, and its driving me
nuts! (game we play...)

Let f be an entire function such that Re(f(z))+Im(f(z)) < 1 for all z.
Show that f is constant.

So far, I've got that f is holomorphic and entire...

Thanks for any help!

.

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