  4. Suppose that (X, dx) and (Y, dy) are metric spaces, and define distance on X × Y byProve that (Tmyn) → (z, y) in X × Y if and only ifFm → z in X and Un → y in Y.5. If f:X → Y, the graph of f is the set G = {(x, f(x)) : x E X}X × Y. Suppose that f is a mappingof a compact metric space X into a metric space Y. Prove that f is continuous on X if and only if itsgraph G is compact. Hint: Use Exercise 4, and for the - part, also Exercise 3 applied in G.

Question help_outlineImage Transcriptionclose4. Suppose that (X, dx) and (Y, dy) are metric spaces, and define distance on X × Y by Prove that (Tmyn) → (z, y) in X × Y if and only ifFm → z in X and Un → y in Y. 5. If f:X → Y, the graph of f is the set G = {(x, f(x)) : x E X} X × Y. Suppose that f is a mapping of a compact metric space X into a metric space Y. Prove that f is continuous on X if and only if its graph G is compact. Hint: Use Exercise 4, and for the - part, also Exercise 3 applied in G. fullscreen
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Step 2

Problem 4:

Given that (X, dX) and (Y, dY) are the metric spaces and the distance is defined as follows.

dX x Y ((x1 , y1), (x2 , y2)) = dX  (x1 , x2) + dY (y1 , y2)

Step 3

To prove that (xn , yn) tends to (x, y) in X x Y if and only if xn tends to x inand yn t...

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