If a ≥ b ≥ c &gt; 0 be real numbers such that ∀ n ∈ N, there exists triangles of side lengths a n , b n , c n . Then prove that those triangles must be Isosceles.

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If   a  ≥  b  ≥  c  &amp;gt;  0 be real numbers such that   ∀  n  ∈  N, there exists triangles of side lengths       a    n    ,      b    n    ,      c    n  . Then prove that those triangles must be Isosceles.

Kiana Cantu · Accepted Answer

Suppose the triangle is not isosceles. Then   a  &amp;gt;  b  &amp;gt;  c  &amp;gt;  0.We need to show that for sufficiently large   n,       a    n    &amp;gt;      b    n    +      c    n  , which violates the triangle inequality.Note that                     a        n                              b          n                +                  c          n                      ≥                    a        n                    2                  b          n                      =            1      2                  (                        a          b                    )        n    →  ∞ as             a      b        &amp;gt;  1.Thus there exists some large   N such that the quotient is larger than 1, and this completes the proof by contrapositive.

If a ≥ b ≥ c > 0 be real numbers such that <

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