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| infinite list of numbers, written in an order |
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| a number is the limit of an increasing sequence {an} if, given any integer k>0, all the an after some place in the sequence agree with L to k decimal places |
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| There exists a number B such that an<=B for all n |
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| A positive increasing sequence which is bounded above has a limit |
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| There exists a number C such that an>=C for all n |
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| A positive decreasing sequence has a limit |
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| increasing for all n or decreasing for all n |
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| a bounded monotone sequence has a limit |
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| lower/upper/sharper estimates |
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| with K<K'<c<M'<M, K is a lower estimate, M is an upper estimate, K' is a sharper lower estimate, and M' is a sharper upper estimate for c |
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| difference forms of triangle inequality |
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abs(a-b)>=abs(a)-abs(b)
abs(a+b)>=abs(a)-abs(b) |
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For any two real numbers a<b, there is
(i) a rational number r(e)Q between them: a<r<b
(ii) an irrational number s(/e)Q between them: a<s<b |
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| {an} has a property P for n large if there is a number N such that an has property P for all n>=N |
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| general completeness property |
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| A sequence which is bounded and monotone for n>>1 has a limit |
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