Show that if (Sn.tn) converges to (s,t) in R² with the do ("max") metric, then the sequence converges to the same limit in the usual d₂ metric.

2) Calculate each of the limits by considering the logarithm of the limit first. a) lim x'n x x-0+ b) lim xsin x x→0+

3. Let t be a real number. Prove tn lim = 0. n→∞ n!

3. Assume that ƒ is uniformly continuous on a bounded set S, prove that f(S) is bounded. (hint: thm 19.4, 11.5) 4. Let f(x) = (x-1)(x-3)2, determine limx→1 ƒ(x), limx→2 ƒ (x), limx→3 ƒ(x). (hint: limit may be infinite or does not exist).

1. Let f : (0,1) → R be uniformly continuous. Let (xn) be a sequence in (0, 1). Suppose that lim,, 0 Tn = 0. Prove that lim, >∞ f (Tn) exists.