(related to Proposition: Difference of Convergent Real Sequences)
The proposition follows immediately from the corresponding proposition for the sum of convergent real series, the definition of subtracting real numbers, and the product of a real number and a convergent real sequence:
\[\begin{array}{rcll}\lim_{n\rightarrow\infty} (a_n - b_n)&=&\lim_{n\rightarrow\infty} (a_n + ( - b_n))&\text{by definition of subtracting real numbers}\\ &=&\lim_{n\rightarrow\infty} a_n + \lim_{n\rightarrow\infty} ( -1 )\cdot b_n&\text{sum of convergent real series}\\ &=&\lim_{n\rightarrow\infty} a_n + (-1)\cdot\lim_{n\rightarrow\infty} b_n&\text{product of a real number and a convergent real sequence}\\ &=&\lim_{n\rightarrow\infty} a_n - \lim_{n\rightarrow\infty} b_n&\text{by definition of subtracting real numbers}. \end{array}\]
