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Proof : Absolute Values

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Letting x,y be elements of the real numbers, prove that |xy|=|x||y|.

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An absolute value proof is provided.

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uniform convergence and proof

Determine the uniform convergence and convergence of the series ∑▒〖〖(f〗_n),〗 where f_n (x)is given by the following: (The Weoerstrass M-Test will be needed)
a sin⁡(x/n^2 ) b. 〖(nx)〗^(-2),x≠0,
c. 〖(x^2+n^2)〗^(-1) d. (-1)^n (n+x)^(-1),x≥0,
e. 〖(x^n+1)〗^(-1),x≥0 f. x^n (x^n+1)^(-1),x≥0.

Suppose (k_n ) is a decreasing sequence of positive numbers. If ∑▒(k_n sin⁡nx ) is uniformly convergent, then lim⁡(nk_n )=0.

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