REVIEW SHEET

• order, ordered set
• bounded above, upper bound
• least upper bound
• field
• ordered field
• one-to-one, onto, bijection
• finite, infinite, countable, uncountable
• equivalence relation
• metric space
• N_r(x)
• open set
• limit point
• closed set
• closure of a set
• interior of a set
• bounded set
• open cover
• compact set
• discrete metric
• convergent sequence
• Cauchy sequence
• convergent series
• f:X-->Y is continuous at p
• f:X-->Y, A subset of X -- define f(A)
• f:X-->Y, B subset of Y -- define f^{-1}(B)
• lim_{x-->p} f(x) = L
• differentiable
• local max/min

• Square root of two is irrational.
• x+y = x+z implies y=z.
• 0x=0
• x>0, y < z implies xy < xz
• Z is unbounded in R (without assuming the Archidean principle)
• sup (0,1) = 1
• N_r(x) is an open set.
• U is open if and only if U^c is closed.
• Arbitrary unions of opens are open.
• Finite intersections of opens are open.
• Analogous statements for closed sets.
• F closed and E a subset of F implies E is contained in F.
• Proving particular sequences converge directly from the definition (e.g. 1/n → 0, 1/n² → 0, ...).
• Limits are unique.
• If E is closed, p_n in E for all n and p_n → p, then p is in E.
• If {s_n} → s and {t_n} → t, then {s_n+t_n} → s+t.
• Compact implies closed.
• Compact implies bounded.
• A descending sequences of non-empty compact sets has a non-empty intersection.
• If K is compact and covered by an ascending sequence of open sets, then K is covered by a single open set.
• Convergent implies Cauchy.
• A closed subset of a compact set is compact.
• Cauchy sequences are bounded.
• K compact and infinite implies that K has a limit point.
• If Σ a_n is convergent, then a_n tends to 0.
• Comparison test (Theorem 3.25)
• f,g continuous implies f+g is continuous
• f:X-->Y continuous, U open in Y implies f^{-1}(U) is open in X.
• f:X-->Y continuous, K compact in X implies f(K) compact
• Extreme value theorem
• differentiable implies continuous
• (f+g)' = f' + g'
• product rule
• local min/max at p implies f'(p)=0
• Rolle's theorem -- that is, f(a)=f(b), f differentiable implies f'(c)=0 for c between a and b
• Mean value theorem
• f'=0 implies f is constant
• f'>0 implies f is increasing
• f'<0 implies f is decreasing