Epsilon Neighborhood

Front 1

Epsilon neighborhood

Back 1

If epsilon is greater than zero, then the interval x minus epsilon, x plus epsilon is an epsilon neighborhood of x

Hint 1

If ϵ > 0 then the interval (x − ϵ, x + ϵ) is called an epsilon neighborhood of x.

Front 2

Neighborhood of x

Back 2

A set Q of real numbers is said to be a neighborhood of x, if there exists an epsilon greater than zero such that x minus epsilon, x plus epsilon is a subset of Q

Hint 2

A set Q of real numbers is said to be a neighborhood of x, if there exists an ϵ > 0 such that (x − ϵ, x + ϵ) ⊂ Q.

Front 3

Sequence said to be Cauchy

Back 3

A sequence A N from n=1 to infinity is said to be Kauchy if and only if for for all epsilon greater than zero there is a positive integer cap N such that for all n greater than or equal to cap N and all m greater than or equal to cap N, we have absolute value of A N minus A M is less than epsilon.

Hint 3

A sequence {an}∞n=1 is said to be Cauchy iff for ∀ϵ > 0 there is a positive integer N such that for alln ≥ N and all m ≥ N we have |an − am| < ϵ.

Front 4

Accumulation Point

Back 4

Let S be a set of real numbers. A real number A is said to be an accumulation point of S if and only if every epsilon neighborhood of A contains infinitely many points of S.

Front 5

Bolzano Weierstrass Theorem

Back 5

Every bounded infinite set of real numbers has at least one accumulation point.

Front 6

Theorem Cauchy

Back 6

Every Cauchy sequence converges to a real number

Note: Every Cauchy sequence is bounded.

Front 7

Theorem

Back 7

Let A N from n=1to infinity be a sequence, it is convergent if and only if it is Cauchy.

Hint 7

Theorem: Let {an}∞ n=1 be a sequence, it is convergent iff it is Cauchy.

Front 8

Squeeze theorem

Back 8

Hint

Hint 8

“Squeeze” Theorem:

Suppose that {an}∞ n=1, {bn}∞ n=1, and {cn}∞ n=1 are three sequences of real numbers such that ∀n an ≤ bn ≤ cn, and such that limn→∞ an = limn→∞ cn = A. Then limn→∞ bn = A.

Front 9

Equivalence statement part A

Back 9

Every epsilon neighborhood of x contains infinitely many points of S.

Front 10

Equivalence statement part B

Back 10

Every neighborhood of x contains infinitely many points of S.

Front 11

Equivalence statement part C

Back 11

Every neighborhood of x contains a point of S that is different from x.

Front 12

Equivalence statement part D

Back 12

Every epsilon neighborhood of x contains a point of S that is different from x.

Front 13

Equivalence statement part E

Back 13

There exists a sequence X of n from n=1to infinity such that for all n (∀n∈N) we have x ̸= x n ∈ S and limn→∞ xn = x.