Final answer:
To prove that an entire function bounded by a constant is constant throughout the complex plane, one can apply Liouville's Theorem, which draws from the Cauchy integral formula and estimates for an entire function's derivatives.
Step-by-step explanation:
The question involves proving that a function f, which is entire and bounded, is a constant function. This is a standard result in complex analysis known as Liouville's Theorem.
According to the theorem, if an entire function is bounded by a constant m for all z in the complex plane, then f must indeed be a constant function.
The proof typically uses the Cauchy estimates for derivatives of an entire function, which result from the Cauchy integral formula, showing that all derivatives vanish if an entire function is bounded. Consequently, this implies that the function has no variation and is hence constant.