Math 149s: Analysis Cheat Sheet
Matthew Rognlie
October 7, 2009
1 Definitions
If Sis some set of real numbers:
1. sup Sis the least upper bound of S.
2. inf Sis the greatest lower bound of S.
Smay or may not contain its sup or inf; if it does, we say that the sup is its maximum and the inf
is its minimum.
For a sequence {an}∞
n=1, we also define:
1. lim sup an= infksupn{an}∞
n=k
2. lim inf an= supkinf n{an}∞
n=k
We can define the limit limanof a sequence in two equivalent ways:
1. The limit is defined if the lim inf and lim sup of the sequence exist and have the same value,
in which case lim an= lim inf an= lim sup an.
2. lim an=cif for any > 0, we can find some Nsuch that for all n≥N,|an−c|< .
lim an=∞if for any y∈Rthere is Nsuch that for all n≥N,an> y.
We say that an infinite series P∞
n=1 bnconverges if the limit of its partial sums limk→∞ Pk
n=1 bn
converges as a sequence.
There are also two equivalent notions of the limit of a function f(x) as x→y:
1. limx→yf(x) = cif for all sequences xn→y,f(xn)→c.
2. limx→yf(x) = cif for every > 0, we can find some δ > 0 such that for all xsuch that
|x−y|< δ,|f(x)−c|< .
A function fis continuous at point yif limx→yf(x) = f(y). Using our two definitions of limits,
we can write this as:
1. fis continuous at yif for any sequence xn→y,f(xn)→f(y).
2. fis continuous at yif for any > 0, we can find some δ > 0 such that for all xsuch that
|x−y|< δ,|f(x)−f(y)|< .
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