Corollary to Theorem 4-4
If a triangle is equiangular, then the triangle is equilateral.
The Corollary to Theorem 4-4 states that if a sequence of real numbers is convergent, then it is bounded.
To understand this statement, we first need to define what it means for a sequence to be convergent. A sequence of real numbers is said to be convergent if there exists a real number L such that for any positive number ε, there exists an index N such that for all n ≥ N, |an − L| < ε. In simple terms, this means that as n approaches infinity, the terms of the sequence get closer and closer to a fixed value L. Now, let's consider what it means for a sequence to be bounded. A sequence of real numbers is said to be bounded if there exists a real number M such that |an| ≤ M for all n. In other words, the terms of the sequence do not grow too large or too small. The Corollary to Theorem 4-4 simply states that any sequence of real numbers that converges, must also be bounded. This makes intuitive sense because if the terms of the sequence were unbounded, then they would grow infinitely large or infinitely small, which would prevent the sequence from converging towards a fixed value. To prove this corollary, we can use a proof by contradiction. Suppose that there exists a sequence of real numbers {an} that is convergent, but is not bounded. Then for any real number M, there exists an index N such that |an| > M for some n ≥ N. However, because the sequence is convergent, we can choose ε = 1 and there must exist an index N1 such that for all n ≥ N1, |an − L| < 1. Now, if we choose M = |L| + 1, we have a contradiction because there exists an index N2 such that |an| ≤ M for all n ≥ N2. This contradiction proves that a sequence of real numbers that is convergent must also be bounded, which is the statement of the Corollary to Theorem 4-4.
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