推 s60984:推證明~~~ 02/17 23:17
推 ikikiki:謝謝 02/17 23:51
※ 引述《ikikiki (小優)》之銘言：
: (a) Let K be a compact set in R^3 and define
: C={(x,y) in R^2 | there exists z in R such that (x,y,z) in K}
: Prove that C is a compact set in R^2.
proof:
Let ((x_n, y_n)) is a sequence of C, then there exists a sequence (z_n) in R
such that (x_n, y_n, z_n) belongs to K for all n.
Since K is compact, then there exists a subsequence (x_n_k, y_n_k, z_n_k)
comverging to (x,y,z) belonging to K.
Since (x,y,z) lies in K, then (x,y) lies in C, so that the subsequence
((x_n_k, y_n_k)) of ((x_n, y_n)) converging to (x,y) in C.
Hence K is compact.
: (b) Let A be a path connected subset of R^n and f:A→R^m be a continuous
: function. Define the graph of f by
: G={(u,v) in R^n+m | u in A , v=f(u)}
: Prove that G is path connected.
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