Analyse I (anglais)

List of proofs (order by the chapter in the notes in which the result appears; we shall indicate also the corresponding numbering in the notes):

• Chapter 1: Proofs:
  -- irrationality of the square root of a prime number p [Proposition 1.1 and exercise 1.a in the exercise sheet for Week 2].
  
  
• Chapter 2: Basic notions and number sets:
  -- existence of the infimum (resp. The supremum) for a set which is bounded from below (reps. from above) [Corollary 2.26];
  -- any set of the natural numbers admits minimum [Proposition 2.34];
  -- density of the rational numbers among the real numbers [Proposition 2.44].
  
  
• Chapter 3: Complex numbers:
  None.
  
  
• Chapter 4: Sequences:
  -- uniqueness of the limit [Proposition 4.22];
  -- a converging sequence is bounded [Proposition 4.27];
  -- addition rule for finite limits [Proposition 4.30];
  -- limit of the product of a bounded sequence with a sequence converging to 0 is also 0 [Corollary 4.41];

• Chapter 5: Series:
  -- the harmonic series does not converge [Example 5.9];
  -- squeeze theorem for sequences [Theorem 5.12];
  -- absolute convergence implies convergence [Proposition 5.28];
  -- Cauchy's criterion [Proposition 5.31].

• Chapter 6: Functions, limits continuity:
  -- composition of continuous functions is continuous [Proposition 6.43];
  -- right and left limits always exist for monotone functions [Proposition 6.60];
  -- Banach's fixed point theorem [Corollary 6.84].
  

• Chapter 7: Differentiability:
  -- points of local extremum are stationary points [Proposition 7.52];
  -- Rolle's theorem [Theorem 7.56];