Show that there exists no sequence of functions satisfying the following

• poet_3000
In summary, the conversation discusses a topology exercise that involves showing the non-existence of a sequence of continuous functions from the set of real numbers to itself that would make the sequence bounded only when the input is rational. The solution involves using the fact that every real number can be approximated by a sequence of rational numbers and showing that the set where the sequence is unbounded is not a G_delta set.
poet_3000
I found this interesting exercise on a topology book I'm reading, but I don't have a clue what to do.

Show that there is no sequence {g_n} of continuous functions from R to R such that the sequence {(g_n)(x)} is bounded iff x is rational (where R = set of real numbers).

poet_3000 said:
I found this interesting exercise on a topology book I'm reading, but I don't have a clue what to do.

Show that there is no sequence {g_n} of continuous functions from R to R such that the sequence {(g_n)(x)} is bounded iff x is rational (where R = set of real numbers).

Every real is the limit of a sequence of rational. The value of a continuous function on this sequence of rationals converges to its value on the limit.

Define $$B_n=\{x\in\mathbb{R}:|f_k(x)|>n\:\mathrm{ for some }\:k\}$$ and note that each $$B_n$$ is open. Now the set of points where the sequence is unbounded is $$\bigcap_{n\ge0}B_n$$. This is a $$G_\delta$$ set, and the rationals are not a $$G_\delta$$ set.

http://en.wikipedia.org/wiki/Gδ_set

Tinyboss,

You are a clever, clever man. How did you think of something like that?

I would approach this problem by first understanding the definitions and concepts involved. In this case, we are dealing with sequences of functions, continuity, boundedness, and rational numbers.

To show that there exists no sequence of functions satisfying the given condition, we need to prove that it is not possible to find a sequence of functions that meets the given criteria. In other words, we need to show that there is a logical contradiction in the statement.

One approach to proving this is through proof by contradiction. Suppose there exists a sequence of continuous functions {g_n} from R to R such that {(g_n)(x)} is bounded if and only if x is rational. This means that for any rational number x, the sequence {(g_n)(x)} is bounded, and for any irrational number x, the sequence is unbounded.

However, this leads to a contradiction since the set of rational numbers and the set of irrational numbers are both dense in R. This means that between any two real numbers, there exists both a rational and an irrational number. Therefore, for any real number x, the sequence {(g_n)(x)} must be both bounded and unbounded, which is impossible.

Thus, we have shown that our initial assumption of the existence of such a sequence leads to a contradiction. Therefore, there exists no sequence of functions {g_n} that satisfies the given condition. This proves the statement and completes the exercise.

What does it mean for a sequence of functions to satisfy a given condition?

A sequence of functions satisfying a given condition means that for every value of the independent variable, the limit of the sequence of functions at that point will satisfy the given condition.

What is a counterexample in mathematics?

A counterexample in mathematics is an example that disproves a statement or concept. It is a specific instance where the given statement or concept does not hold true.

What does it mean to show that there exists no sequence of functions satisfying a given condition?

To show that there exists no sequence of functions satisfying a given condition means to prove that there is no possible sequence of functions that can meet the given condition for all values of the independent variable.

Why is it important to prove that there exists no sequence of functions satisfying a given condition?

Proving that there exists no sequence of functions satisfying a given condition is important because it allows us to identify the limitations and boundaries of a certain concept or statement. It also helps us to understand the behavior and properties of functions in a more precise and rigorous manner.

What are the steps involved in proving that there exists no sequence of functions satisfying a given condition?

The steps involved in proving that there exists no sequence of functions satisfying a given condition may vary depending on the specific condition, but generally involve using logical reasoning and mathematical techniques such as proof by contradiction or direct proof. It also requires a thorough understanding of the given condition and the properties of functions.

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