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| The Triangle Inequality [If a and b are any two real numbers then Abs(a+b)<= Abs(a) + Abs(b)] [Various forms. Most important inequality in mathematics.] |
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Definition
Proof There are four possibilities:
(a) If a 0 and b 0, then a C b 0, so ja C bj D a C b D jaj C jbj.
(b) If a 0 and b 0, then a C b 0, so ja C bj D |
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| How prove convergence of sequence S_n to s |
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Definition
| Show that forAll epsilon>0, \exists{N} s.t. n>N \implies \abs{s_n-s} < epsilon |
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Definition
| Suppose a_1, ..., a_n are integers and r is a rational# satisfying polynomial eqn: (a_n)x^N+...+(a_o)x^0 where n\geq 1, a_n !=0 \and a_0 !=0. Write r=p/q where p,q are relatively prime integers then q divides a_n and p divides a_0. |
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Definition
| # that satisfies polynomial eqn: (a_n)x^n+...+(a_0)x_0, where the coeffs a_i \in \Z, a_n != 0 and n\geq 1 |
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Definition
If S \ProperSubset N satisfies
(a) 1 \element S
(b) n /element S \implies n+1 \element S
then S=N |
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Definition
|a|=a if a \geq 0
|a|=-a if a < 0 |
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Definition
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| Lec2 'More' Triangle Thms |
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Definition
|a-b| \leq |a|+|b|
||a|-|b|| \leq |a+b|
||a|-|b|| \leq |a-b| |
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Definition
| Every non-empty subset S of IR that has an upper bound in IR has a least upper bound, or supremum, in IR . |
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Definition
| Let S be a nonempty subset of IR . If \exists s_0 \in S s.t. s\leq s_o \ForAll s\in S, then s_0 is called the maximum of S. (s_0=maxS) |
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Definition
| Let S be a nonempty subset of IR . If \exists s_0 \in S s.t. s_0\leq s \ForAll s\in S, then s_0 is called the minimum of S. (s_0=minS) |
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| upper bound, bounded above (Let S be a nonempty subset of IR) |
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Definition
| If \exists M\in IR s.t. s\leq M \ForAll s\in S, then M is an upper bound for S and S is said to be bounded above. |
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| lower bound, bounded below (Let S be a nonempty subset of IR) |
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Definition
| If \exists m\in IR s.t. m\leq s \ForAll s\in S, then m is an upper bound for S and S is said to be bounded below. |
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Definition
| If S is bounded above and below, it said to be bounded |
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| sup(S), inf(S) [Let S \ProperSubset IR, S != \null] |
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Definition
Define
U={u \in IR: s\leq u \ForAll s\inS}
L={u\in IR: u\leq s \ForAll s\in S}.
Then minU if it exists is the sup(S)
maxL if it exists is the inf(S) |
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| A sequence (s_n) of real numbers is called a nondecreasing sequence iff |
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Definition
| s_n\leq s_{n+1} \ForAll n |
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Definition
| If a>0 and b>0, then for some pos. integer n, we have na>b |
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Definition
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| existence of max(S), min(S), sup(S), inf(S) |
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Definition
| max and min of S are elements of S and need not exists. Sup and inf need not be elements of S and exist in IR. |
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| \Q is dense in IR (Thm 4.7) |
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Definition
| If a,c\in IR; If a,c\in IR with a |
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Definition
| Let \inf and -\inf be any two fixed objects s.t.; -\inf != \inf; \inf !\in IR , -\inf !\in IR; then R^# = IR\Union {\inf} \Union {-\inf} is called the extended real number system |
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Definition
Let S \ProperSubset IR, s != \null then;
if S has no upper bound in IR
define sup(S) = \inf
if S has no lower bound in IR
define inf(S) = -\inf |
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Definition
| A sqn is a function S(n): \N \rightarrow A |
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| A sequence of real#s, (s_n), converges to L \in IR iff ___. A sqn diverges when ___. |
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Definition
\ForAll \epsilon>0 \exists N\in IR s.t. n>N \implies |s_n-L|<\epsilon
If sqn does not converge, we say it diverges
in this case, lim_{n\to\inf}=L |
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Definition
| Let (a_n), (b_n), and (s_n) be sqns in IR with A_n\leq s_n\leq b_n \ForAll n\in \N. And lim{a_n}=lim{b_n}=s, then lim{s_n}=s |
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| A sequence is bounded iff [Def 9.(-1)] |
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Definition
| A sqn (s_n) _{ n\in\amalg} is bounded iff the set {s_n \bar n\in\amalg} is a bounded set. |
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| A sqn is bounded iff [prop 9.0 and hw] |
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Definition
| A sqn (s_n) is bounded iff \exists M>0 s.t. |s_n|\leq M \ForAll n |
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| The sequential limit laws |
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Definition
LET (s_n) AND (t_n) be sqns which converge to s!=0 and t!=0, then: lim(k*s_n)k*lim(sn)
lim(s_n+t_n)=lim(s_n)+lim(t_n)
lim(s_n*t_n)=st=lim(s_n)lim(t_n)
\exists N\in IR s.t. n>N\implies |s_n|>|s|2 [Lemma 9.5(a)]
lim(1/s_n)=1/s=1/lim(s_n) [hw]
lim(s_n/t_n)=s/t=lim(s_n)/lim(t_n) |
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| When does sequence (s_n) diverge to -\inf? |
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Definition
| We say (s_n) diverges to -\inf [that is, lim(s_n)=-\inf] iff; \ForAll M<0, \exists N\in\N s.t.; n>N\implies s_n |
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| When does sequence (s_n) diverge to +\inf? |
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Definition
| (s_n) diverges to +\inf [that is, lim(s_n)=+\inf] iff; \ForAll M>0, \exists N\in\N s.t.; n>N\implies s_n>M. |
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| A sequence (s_n) of real numbers is called a Cauchy sequence iff |
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Definition
| \ForAll \epsilon>0\exists N s.t. m,n>N\implies |s_n-s_m|< \epsilon |
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| All convergent sequences are |
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Definition
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Definition
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| A sequence (s_n) of real numbers is called a incecreasing sequence iff |
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Definition
| s_n\geq s_{n+1} \ForAll n |
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Definition
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Definition
| If (s_n) is either nondecreasing or nonincreasing, we call it a monotone (or monotonic) sqn |
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| Thm 10.7 Let (s_n) be a sequence in IR |
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Definition
i) If lim(s_n) is defined, then liminf(s_n)=lim(s_n)=limsup(s_n)
ii) If liminf(s_n)=limsup(s_n), then lim(s_n)=liminf(s_n)=limsup(s_n) |
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Definition
| Every nonempty subset of IR bounded below has an infimum. |
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