The limits on an as n goes to infinity is a
Then for some epsilon greater than 0, chose N such that for n>N
we have |an-a| < epsilon.
Now if m and n are > N we have |an-am|=|(am -a)-(an -a)|
< or= |am -an | which is < or equal to 2 epsilor so the sequence is Cauchy.
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Every convergent sequence is Cauchy. Every Cauchy sequence in Rk is convergent, but this is not true in general, for example within S= {x:x€R, x>0} the Cauchy sequence (1/n) has no limit in s since 0 is not a member of S.
That's the famous Fibonacci sequence, where every term is the sum of the previous two.
in math ,algebra, arithmetic
Every next number is increased by 5. The next number in the sequence is 23.
Mathematical Induction is a process uses in College Algebra It can be used to prove that a sequence is equal to an equation For Example: 1+3+5+7+n+2=2n+1 there are 3 steps to mathematical induction the first includes proving that the equation is true for n=1 the second includes substituting k for every n-term the third involves substituting k+1 for every k-term to prove that both sides are equal