Consider two sequences $\{a_n\}_{n=1}^\infty$ and $\{b_n\}_{n=1}^\infty$ where $a_n\geq 0 \forall\ n$ and $b_n\geq 0\ \forall n$. Is it true that $$\limsup_{n\rightarrow\infty}\frac{1}{n} \log\left(e^{n*a_n}+e^{n*b_n}\right) = \max\left\{\limsup_{n\rightarrow\infty} a_n , \limsup_{n\rightarrow\infty} b_n\right\}$$

My attempt is as follows: Let us denote $a=\limsup_{n\rightarrow\infty} a_n$ and $b= \limsup_{n\rightarrow\infty} b_n$. Since $$f(x,y,z)= \frac{1}{x} \log\left(e^{x*y}+e^{x*z}\right)$$ is continuous in all the three parameters we have $$ \begin{eqnarray} \limsup_{n\rightarrow\infty}\frac{1}{n} \log\left(e^{n*a_n}+e^{n*b_n}\right) &=& \limsup_{n\rightarrow\infty}f(n,a_n,b_n)\\ &=& \limsup_{n\rightarrow\infty}\lim_{a_n\rightarrow a}\lim_{b_n\rightarrow b}f(n,a_n,b_n)\\ &=& \limsup_{n\rightarrow\infty} f(n,a,b)\\ &=& \max(a,b) \end{eqnarray} $$


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