- #1
mancini0
- 31
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Homework Statement
Hi everyone. I must show that if f is a continuous function over the complex plane, with
limit as z tends to infinity = 0, then f is in fact bounded.
The Attempt at a Solution
Since f is continuous and lim z --> infinity f(z) = 0, by definition of limit at infinity I know
for all epsilon > 0, there exists N > 0 such that for all |z| > N, |f(z)-0| < epsilon.
The problem gives the hint that I would be wise to consider epsilon = 1. With that I see that
|f(z)| < 1 for all |z| > N.
Then |f(z)| is bounded. I need to extend this to f(z). I know from absolute convergence that if the absolute value of a series converges, the series itself converges. But we are not dealing with series, nor have we yet learned about series in the complex plane.
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