Problem on sequences and limits

Question

Assignment:

Q1.  If (b_n) is a bounded sequence and lim(a_n) = 0, show that the lim(a_nb_n) = 0.

a) Let X = (x_n) and Y = (y_n) be sequences of real numbers that converge to x and y respectively, and let c be an element R.  Then the sequences X+Y, X-Y, X·Y, and cX converge to x+y, x-y, xy, and cx, respectively.

b) If X = (x_n) converges to x and Z = (z_n) is a sequence of nonzero real numbers that converges to z and if z ≠ 0, then the quotient sequence X/Z  converges to x/z.

Q2. Let x₁ > 1 and x_n+1 : = 2 - 1/x_n (for n an element of N).  Show that (x_n) is bounded and monotone.  Find the limit.

Provide complete and step by step solution for the question and show calculations and use formulas.