Is Uniform Convergence Implying Boundedness of the Limit Function?

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Uniform convergence of a sequence of functions (fn) to a limit function f does not necessarily imply that f is bounded, even if each fn is bounded. The key point is that uniform convergence ensures that the difference between fn and f can be made arbitrarily small, but it does not restrict the behavior of f itself. A counterexample can illustrate this: if fn converges uniformly to a function that diverges, then f can be unbounded despite all fn being bounded. Thus, uniform convergence does not guarantee the boundedness of the limit function. Understanding this relationship is crucial in analysis.
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Homework Statement



Just trying to get feel for uniform convergence and it's relationship to boundedness. If a sequence of functions (fn) converges uniformly to f and (fn) is a sequence of bounded functions, is f also bounded?

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The Attempt at a Solution

 
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If the convergence is uniform then for all e there is an N such that |fn(x)-f(x)|<e for all n>=N. If fN is bounded and f unbounded, how can this be?
 
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