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Math Help - Can anyone help me prove a logic statement about limits?

  1. #1
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    Can anyone help me prove a logic statement about limits?

    Hi I have this assignment and I am having trouble on going about the proof structure and the process of proving it.

    first of all here's the question:

    PROVE or DISPROVE the statement:
    ∀ x ∈ ℝ, ∀ ε ∈ ℝ, ∃ δ ∈ ℝ, ∀ y ∈ ℝ, |x - y|< δ ⇒ |x^2 - y^2| < ε

    I'm very confused on how to do this. I know that this is the limit definition as well but I don't know how to go about proving this statement. So can anyone please help me figure this out? Help is much appreciated. Plus can anyone give me some tips on how to tackle statements like this and tips on how to decide whether the statement is to proven or to be disproved. Thank you in advance.
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  2. #2
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    Quote Originally Posted by gello88 View Post
    Hi I have this assignment and I am having trouble on going about the proof structure and the process of proving it.

    first of all here's the question:

    PROVE or DISPROVE the statement:
    ∀ x ∈ ℝ, ∀ ε ∈ ℝ, ∃ δ ∈ ℝ, ∀ y ∈ ℝ, |x - y|< δ ⇒ |x^2 - y^2| <
    Is statement about limits? I'm a bit ignorant but I thought the limit was defined as:

    A function f has limit L at a, if ∀ε > 0 ∃δ > 0: ∀x (0 < |x - a| < δ ⇒ |f (x) - L| < ε)

    Your statement seems to say that for the real number x and real number δ there exists a real number such that for each y if the absolute value of the difference between x and y is less than δ, then the absolute value of the squares of x and y is less than....
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  3. #3
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    This is the definition of a continuous function applied to the function f(x) = x^2. I remember that when I first confronted epsilon-delta definitions I was also confused. It takes time to settle this in one's head and to form the needed connections in the brain.

    The formula basically says the following. Consider the graph of f(x) = x^2, i.e., a parabola. Let's assume the horizontal axis is called x and the vertical one is called z, since y is already used in the definitions. Suppose x and \epsilon are given. Take any x and mark x^2 on the z axis. Then draw an interval of length 2\epsilon with the center at x^2 on the z axis. The formula claims that there is, in turn, an interval around x on the x axis that is completely mapped into the interval around x^2 on the z axis.

    In other words, by shrinking the interval around x we can arbitrarily shrink the length of the image of this interval. I.e., if x changes a little bit, x^2 will also change a little; we can make this change of x^2 as small as we want.

    You can see my post on quantifiers here.
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