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Prove by induction on n that for all n ∈ N, there exists k ∈ N such that k2 ≤ n...

Feb 20, 2024

Prove by induction on n that for all n ∈ N, there exists k ∈ N such that k2 ≤ n < (k + 1)2. (In your proof, you may use the fact that 0 ≤ x < y implies x2 < y2.

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Solution by Steps

step 1

To prove by induction, we start with the base case, where

n = 1

step 2

For

n = 1

, we can choose

k = 1

since 1^2 \leq 1 < (1 + 1)^2

step 3

Assume the statement is true for some

n

, i.e., there exists a

k

such that k^2 \leq n < (k + 1)^2

step 4

Consider

n + 1

. If n + 1 < (k + 1)^2 , then k^2 \leq n < n + 1 < (k + 1)^2 , and the statement holds for

n + 1

with the same

k

step 5

n + 1 = (k + 1)^2

, then the statement holds for

n + 1

with

k + 1

since (k + 1)^2 \leq n + 1 < (k + 2)^2

step 6

By the principle of mathematical induction, the statement is true for all

n \in \mathbb{N}

Answer

The statement is proven by induction and holds for all

n \in \mathbb{N}

Key Concept

Mathematical Induction

Explanation

Mathematical induction is a proof technique that establishes the truth of an infinite sequence of propositions. By proving the base case and the inductive step, we can conclude the statement holds for all natural numbers.