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2.1 Real Vocab
| Term | Definition |
|---|---|
| Absolute Value | For a real number x, the absolute value of x, denoted |x|, is defined by |x| = {x, if x ≥ 0; -x, if x < 0 |
| Triangle Inequality | For all x, y ∈ ℝ, we have |x + y| ≤ |x| + |y| |
| ε-neighborhood | Let p ∈ ℝ, and let ε > 0. The set Nε(p) = {x ∈ ℝ: |x - p| < ε} is called the ε-neighborhood of point p. |
| Euclidean Distance | For x, y ∈ ℝ, the euclidean distance d(x, y), between x and y is defined by d(x, y) = |x - y| |
| Sequence | If A is a set, by a sequence in A we mean a function f from ℕ into A. For each n ∈ ℕ, let Xn = f(n). Then Xn is called the nth term of the sequence f. Denoted by {Xn} from n = 1 to infinity |
| Converge | A sequence {Pn} from n = 1 to infinity ⊆ ℝ is said to converge if there exists p ∈ ℝ such that for all ε > 0 there exists n_0 ∈ ℕ with |Pn - p| < ε for all n >= n_0 We say {Pn} converges to p and we write lim n → ∞ Pn= p or Pn → p |
| Diverge | If {Pn} from n = 1 to infinity does not converge, then it diverges. For all p ∈ ℝ, there exists ε > 0 such that |Pn - p| > ε for all n ∈ ℕ |
| Bounded Sequence | A sequence {Pn} ∈ ℝ is said to be bounded if there exists a positive constant M such that |Pn| ≤ M for all n ∈ ℕ |
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Mollygsmithh
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