67. Weierstrass theorem

The Weierstrass Theorem, also known as the Extreme Value Theorem, is a fundamental concept in mathematics, particularly in real analysis and the analysis of real functions. This theorem is named after Karl Weierstrass, a 19th-century German mathematician who contributed significantly to the theory of analytic functions and number theory.

Weierstrass's theorem states that if a function is continuous on a closed, bounded interval, then that function must have a minimum and a maximum on that interval. In other words, the function reaches its highest and lowest values ​​somewhere in the range. This theorem is one of the cornerstones of calculus and has applications in many areas of mathematics, including analysis, optimization, and geometry.

The proof of Weierstrass's Theorem is an excellent example of how the concepts of continuity and limitation can be used to prove important results. The proof begins by assuming that we have a continuous function f on a closed interval [a, b]. Since the interval is closed and bounded, we can choose a sequence of points x_n in the interval such that the sequence of values ​​f(x_n) is a Cauchy sequence. Since the Cauchy sequences are bounded, the sequence f(x_n) is also bounded.

Next, we use the continuity of f to show that the sequence f(x_n) has a limit on the interval [a, b]. Specifically, since f is continuous, the limit of f(x_n) as n approaches infinity must equal the f of the limit of x_n as n approaches infinity. Since x_n is a sequence in the interval [a, b], its limit must also be in the interval [a, b]. Therefore, the limit of f(x_n) is in the interval [a, b].

Finally, we show that this limit is indeed the minimum and maximum of the function on the interval. To do this, we consider any other point y in the interval [a, b]. Since f is continuous, the function f must be close to f(y) for x close to y. But since the limit of f(x_n) is the minimum and maximum of the function, f(y) must be greater than or equal to the minimum and less than or equal to the maximum. Therefore, the minimum and maximum of the function are indeed reached in the interval.

In summary, Weierstrass' Theorem is a powerful tool in mathematics that allows us to find the extreme values ​​of a continuous function on a closed and bounded interval. This theorem is fundamental to many areas of mathematics and is an essential part of any real analysis or calculus course. For students preparing for the ENEM, understanding this theorem and how it can be applied is crucial to solving complex math problems.

Finally, although Weierstrass's Theorem may seem intimidating at first glance, the proof of the theorem is actually a beautiful demonstration of the interplay between continuity, limitation, and sequences. With practice and study, any math student can learn to understand and apply this important theorem.

Next page of the Free Ebook:

68Heine's Theorem

3 minutes