[SOLVED] Natural Numbers/ Integers Closure of Addition/Multiplication

1. Let m and n be natural number. Prove that the sum and product of natural numbers are natural numbers.

2. Prove that the sum, difference, and product of integers are also integers.

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1. I plan to do this by induction. Let $\displaystyle S(n)=(m+n) \in N $

Base step: Fix m, let n = 1

$\displaystyle S(m+n)=(m+1) \in N$

The problem I have is that isn't this begging the question? I'm trying to prove that the sum of two natural numbers is a natural number, but since m is a natural number how can I just assume that m+1 is a natural number?

2. My attempt: Suppose that m, n,r,s are natural numbers and m > n. Then there exists an integer k such that n+k=m. Thus, k = m-n. Likewise, let l be an integer l = r - s. Again, isn't this begging the question?

Thank you for your time.