# Proof of Convergence and Limits Laws

Let (a_n)^infinity, n=m and (b_n)^infinity, n=m be convergent sequences of real numbers, and let x, y be the real numbers x : = lim_n-->infinity a_n and

y : = lim_n-->infinity b_n

Prove: a) The sequence (a_n b_n) ^ infinity, n=m converges to xy; in other words lim_n-->infinity (a_n b_n)= (lim_n-->infinity a_n) (lim_n-->infinity b_n)

b) Suppose that y does not equal zero, and b_n does not equal zero for all n >= m. Then the sequence (b_n^-1) ^infinity, n=m converges to y^-1, in other words

lim_n-->infinity b_n^-1 = (lim_n-->infinity b_n)^-1

c) The sequence (max(a_n, b_n))^ infinity, n=m converges to max(x,y) in other words, lim_n-->infinity max(a_n, b_n) = max (lim_n-->infinity a_n, lim_n-->infinity b_n)

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#### Solution Summary

Proof of convergence and limits laws are examined.