show directly from the definition that if (x n ) and (y n ) are Cauchy sequences

Question

show directly from the definition that if (xn) and (yn) are Cauchy sequences, then (xn+yn)and (xnyn) are Cauchy sequences.

Explanation / Answer

let x= lim x_n and y = lim y_n . we need to show that e > 0 there is natural number N so that n= N , then ¦( x_n + y_n ) - ( x + y )¦= e . Given any e > 0 we have e/3 > 0 so from the definition of convergence there is a natural number N_x so that ¦x_n - x¦= e/3 for all n > N_x ; similarly we can choose N_y , ¦y_n - y¦= e/3 for all n>N_y Let N = max( N_x , N_y ). if n>N , then by triangle inequality we have ¦( x_n + y_n ) - ( x + y )¦= ¦( x_n - x ) + ( y_n - y )¦ y . Given e > 0 there exists some N_x and N_y such that ¦x_n - x¦ N_x and ¦y_n - y¦ N_y then for every n > max( N_x , N_y ) ¦x_n*y_n - x*y¦= ¦(x_n - x)*y_n + x*(y_n - y)¦ = ¦x_n - x¦*M_y + ¦y_n- y¦*M_x = e / 2 + e / 2 = e

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