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1. Let A ⊆ R and p ∈ R. We say that A is bounded away...

1. Let A ⊆ R and p ∈ R. We say that A is bounded away from p if there is some c ∈ R+ such that |x − p| ≥ c for all x ∈ A. Prove that A is bounded away from p if and only if p not equal to A and the set n { 1 / |x−p| : x ∈ A} is bounded.

2. (a) Let n ∈ natural number(N) , and suppose that k 2 < n < (k + 1)2 for some k ∈ N. Prove that n does not have a square root in N.

(b) Let c ∈ R \ {0}. Prove that if c has a square root in Z, then c has a square root in N.

(c) Prove that 2 does not have a square root in Z

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