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Thank you

let S

Let S be its range. Then S is bounded.

Since R is complete, sup S exists. Let x= sup S

then for all ε > 0, ∃ N

let N = max (N

then d(S

Hence S

and we're done.

let S

_{n}be Cauchy seq in RLet S be its range. Then S is bounded.

Since R is complete, sup S exists. Let x= sup S

then for all ε > 0, ∃ N

_{1}in**N**stx - ε/2 <S

_{N1}<= x⇒ x - ε/2 <S

_{N1}< x + ε/2

so d(S

now for all ε > 0, ∃ N_{N1}, x) < ε/2_{2}in**N**st for all n=>N_{2}implies d(S_{N2}, S_{n})<ε/2let N = max (N

_{1}, N_{2})then d(S

_{n}, x) <= d(S_{N}, S_{n}) + d(S_{N}, x)< εHence S

_{n}converges to x.and we're done.