By Serge Lang
As much as my wisdom, this is often the single e-book that constructs the Lebesgue essential for capabilities to a common Banach-space rather than the true numbers (thus saving us from the pointless and esthetically dissapointing building via confident and destructive functions).I have no idea how Lang does it, yet eerytime you are going to decide up one among his books, you will wonder on the great thing about arithmetic !
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Additional info for Real Analysis
Divisibility and Primes 16. Compute the convergents pn /qn of the simple continued fraction 1, 1, 1, 1, 1, 1, 1 . Observe that pn fn+1 = qn fn for n = 0, 1, . . , 6. 17. Prove that f1 + f2 + · · · + fn = fn+2 − 1 for all positive integers n. 18. Prove that fn+1 fn−1 − fn2 = (−1)n for all positive integers n. 19. Prove that fn = fk+1 fn−k + fk fn−k−1 for all k = 0, 1, . . , n. Equivalently, fn = fn−1 + fn−2 = 2fn−2 + fn−3 = 3fn−3 + 2fn−4 = 5fn−4 + 3fn−5 · · · . 20. Prove that fn divides f n for all positive integers .
Prove that (fn , fn+1 ) = 1 for all nonnegative integers n. In Exercises 16–23, fn denotes the nth Fibonacci number. 24 1. Divisibility and Primes 16. Compute the convergents pn /qn of the simple continued fraction 1, 1, 1, 1, 1, 1, 1 . Observe that pn fn+1 = qn fn for n = 0, 1, . . , 6. 17. Prove that f1 + f2 + · · · + fn = fn+2 − 1 for all positive integers n. 18. Prove that fn+1 fn−1 − fn2 = (−1)n for all positive integers n. 19. Prove that fn = fk+1 fn−k + fk fn−k−1 for all k = 0, 1, . . , n.
Use the sieve of Eratosthenes to ﬁnd the prime numbers up to 210. Compute π(210). 2. Let N = 210. Prove that N − p is prime for every prime p such that N/2 < p < N . Find a prime number q < N/2 such that N − q is composite. 3. Let N = 105. Show that N − 2n is prime whenever 2 ≤ 2n < N . This statement is also true for N = 7, 15, 21, 45, and 75. It is not known whether N = 105 is the largest integer with this property. 4. Let N = 199. Show that N − 2n2 is prime whenever 2n2 < N . It is not known whether N = 199 is the largest integer with this property.
Real Analysis by Serge Lang