Proving infinite intersection has only one point

We have: $0 \le |b_n - a_n|= \dfrac{|b_{n-1} - a_{n-1}|}{\sqrt{b_{n-1}+5}+\sqrt{a_{n-1}+5}}< \dfrac{|b_{n-1} - a_{n-1}|}{2\sqrt{5}}<...< \dfrac{|b_1 - a_1|}{(2\sqrt{5})^{n-1}} = \dfrac{3}{(2\sqrt{5})^{n-1}}\implies \displaystyle \lim_{n \to \infty} |b_n - a_n| = 0$. Further, you can show: $[a_n, b_n] \subseteq [a_{n-1}, b_{n-1}]\implies \bigcap_{n=1}^{\infty}[a_n,b_n]= \{a\}= \{\frac{1+\sqrt{21}}{2}\}$ ( the common limit of both sequences ).