limit of a recursive sequence, Am I allowed to divide by $b_n^2$?

Question

$$b_1 > 0$$ $${b_{n + 1}} = {{{b_n}^2 + 1} \over {{b_n}}} = {{{{{b_n}^2} \over {{b_n}^2}} + {1 \over {{b_n}^2}}} \over {{{{b_n}} \over {{b_n}^2}}}}\mathop = {{1 + {1 \over {{b_n}^2}}} \over {{1 \over {{b_n}^{}}}}} = {1 \over {{1 \over {{b_n}^{}}}}} + {{{1 \over {{b_n}^2}}} \over {{1 \over {{b_n}^{}}}}} = {b_n} + {1 \over {{b_n}^{}}} = + \infty $$

Is it right?