Proof Techniques Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Which proof technique is most appropriate for: 'There exists a real number xx such that x2=2x^2 = 2'? Apply it.

Solution

  1. 1
    The claim is existential ('there exists'), so we only need to find one example — a constructive (direct) proof.
  2. 2
    Exhibit: x=2∈Rx = \sqrt{2} \in \mathbb{R} (since R\mathbb{R} is complete). Then x2=(2)2=2x^2 = (\sqrt{2})^2 = 2. □\square

Answer

x=2 witnesses the existencex = \sqrt{2} \text{ witnesses the existence}
Existential claims are proved by constructing a witness — one object satisfying the property. No induction or contradiction is needed. Choosing the right technique saves effort.

About Proof Techniques

Proof techniques are standard strategies for establishing mathematical claims under different structures.

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More Proof Techniques Examples

Example 1 easy

Name four proof techniques, give a one-sentence description of each, and identify which is best suit

Example 2 medium

Compare direct proof and proof by contrapositive for: 'If [formula] is even, then [formula] is even.

Example 4 medium

Prove using mathematical induction: [formula] for all [formula].

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