n
1. Suppose that the function f:R -> R is continuous and that f(u)≧||u|| for
n -1
every point u in R . Prove that f ([0,1]) is sequentially compact.
2
2. Determine which of the following subset of R is sequentially compact and
2
which of R is path connect.
2 -1
(1) A = {(x,y) 屬於 R : 0 < x ≦ 1, y = (sinx) }
2 -1
(2) B = {(x,y) 屬於 R : 0 < x ≦ 1, y = sin(x ) }
2 x^2 * sinx
(3) C = {(x,y) 屬於 R : 0 ≦ x ≦1, y = -----------------
e^x
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