QuestionQuestion

1. In any metric space, suppose that {xₙ} is a sequence, k is a point and that any subsequence {xₙₖ} has itself a subsequence {xₙₖⱼ} with xₙₖⱼ -> x. Show that xₙ -> x. Hint: by contradiction

2. Show that the limit of a sequence, if any, is unique, i.e. if xₙ -> x and xₙ -> x', then x = x'.

3. In R, suppose xₙ ≤ 3, all n ∈ N and xₙ -> x. Show that x ≤ 3.

4. In N, suppose xₙ -> x. Show that there exists n ∈ N with xₙ = x, all n ≥ N.

5. Suppose d and d and ᶑ are equivalent metrics and on a set X, i.e. (def) There exist m > 0, m' > 0 such that for all x, y ∈ X, d(x, y) ≤ m ᶑ(x,y), ᶑ(x,y) ≤ m' d(x,y). Show
a) E is open in (X, d) <-> E is open in (X, ᶑ)
b) The Euclidean and taxicab metrics and equivalent in R²

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