Limit of a sequence of real numbers If $(a_n), (b_n)$ are two sequences of real numbers so that $(a_n)\rightarrow a,\,\,(b_n)\rightarrow b$ with $a, b\in \mathbb{R}^+$.  How to prove that $a_n^{b_n}\rightarrow a^b$ ?

The function $f(x,y) = x^y = e^{y \ln x}$ is continuous on $\mathbb{R}_+ \times \mathbb{R}$, hence if $(a_n,b_n) \to (a,b)$ (with $a_n, a >0$, of course), then $a_n^{b_n} = f(a_n,b_n) \to f(a,b) = a^b$.
