Homework 4, due on Wednesday, May 6 by 11:59pm PST

• Chapter 8 (8.1, 8.4, 8.5)
• Problems II on p. 115-119 (15)
• Problems IV on p. 225-228 (1, 2, 6, 10)
• Given a sequence \(f\colon\mathbb{Z}^+\to\mathbb{R}\) of real numbers, we say that the sequence converges to a real number \(L\) if for all \(\epsilon>0\), there exists a positive integer \(N\) such that for any positive integer \(n\), if \(n\geq N\), then \(|f(n)-L|<\epsilon\).
  • Prove that the sequence \(f\colon\mathbb{Z}^+\to\mathbb{R}\) where \(f(n)=\frac{n}{n+1}\) converges to \(1\).
  • Prove that if a sequence \(f\) converges to \(L\) and a sequence \(g\) converges to \(M\), then the sequence \(f+g\) converges to \(L+M\). (You may use the triangle inequality: \(|a+b|\leq|a|+|b|\) for any real numbers \(a\) and \(b\).)
• Given a function \(f\colon\mathbb{R}\to\mathbb{R}\) and real numbers \(a\) and \(L\), we say that the limit of \(f\) as \(x\) approaches \(a\) is \(L\) if for all \(\epsilon>0\), there exists \(\delta>0\) such that for all \(x\), if \(0<|x-a|<\delta\), then \(|f(x)-L|<\epsilon\). Prove that if \(f(x)=3x+4\), then the limit of \(f\) as \(x\) approaches \(-1\) is \(1\).