Expert answer:convergent subsequence

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1. Let (X,||.||) be a normed vector space and let W⊆X be a vector subspace. If W
is 1-dimentional, then prove that cl(W)=W. (Hint: use the
Bolzano-Weierstrass Theorem for sequences in R.)
2. Let X be a non-empty set and let d, d’ be distance functions on X. If the
topology of (X,d) coincides with the topology of (X,d’), then prove that d is
equivalent to d’. (d is equivalent to d’ if they have exactly the same sequence
with exactly the same limits.)
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