Let
$\displaystyle \lim\limits_{n\to\infty}q_n=q$
$\displaystyle \lim\limits_{n\to\infty}p_n=p$
Prove that
$\displaystyle \lim\limits_{n\to\infty}min\{q_n,p_n\}=min\{q,p\}$
Suppose q = p. Then every subsequence of q and p converge to q = min{q, p}, hence the minimum of q_n and p_n converges to q. On the other hand, suppose without loss of generality that q < p. Let r = (p + q)/2. Then there exists an N such that for all n >= N, |q - q_n| < r - q. Also, there exists an M such that for all m >= M, |p - p_n| < p - r. Let K = max{M, N}. Then q_k < (p + q)/2 < p_k for all k >= K, hence min{p_k, q_k} = q_k for all k >= K. Therefore the sequence converges to q.