# How can i prove these two convergence theorem?

1. If {an} and {bn} are convergent, then {an士bn} and {anbn} are also convergent

2. If {an} and {bn} are convergent and there exists a constant k>0 such that |bn| > k for all n=1, 2, ...., then {an/bn} is also convergen

CompuChip
$$\forall_{\epsilon > 0} \exists_{N_+} : n > N_+ \implies |(a_n + b_n) - L| < \epsilon$$
where L = a + b is the postulated limit. Of course you already know that given such an $\epsilon$ you can find Na and Nb such that
$$n > N_a \implies |a_n - a| < \epsilon$$
$$n > N_b \implies |b_n - b| < \epsilon$$