Answer
Please see below.
Work Step by Step
We prove the proposition by induction on $n$.
Because of Theorem 1.1 part 2. the statement is clearly true for the case $n=1$. Now suppose that the statement is true for $n=k$, that is,$$\lim_{x \to c}x^k=c^k.$$We must show that this is also true for $n=k+1$. We have$$\lim_{x \to c}x^{k+1}=\lim_{x \to c}x^kx=c^kc=c^{k+1},$$where the last equality was obtained from Theorem 1.2 part 3., that is $\lim_{x \to c}[f(x)(g(x)]=[\lim_{x \to c}f(x)][\lim_{x \to c}g(x)]$, Theorem 1.1 part 2., and the induction hypothesis.
