Let $X$ be a metric space. Suppose $\mathcal{F}$ and $\mathcal{G}$ are families of real valued functions. If $\mathcal{F}$ and $\mathcal{G}$ are equicontinuous, is $\mathcal{F}+\mathcal{G}$ equicontinuous, too?
I think the answer is yes, because the elements of equicontinuous family of functions is continuous and summation of two continuous functions is continuous.
This is my attempt: Let $\varepsilon>0$. Since $\mathcal{F}$ is equicontinuos, then there exists $\delta_1>0$, such that for all $x,y\in X$ with $d(x,y)<\delta_1$,we have $|f(x)-f(y)|<\varepsilon$, for all $f\in \mathcal{F}$.
And since $\mathcal{G}$ is equicontinuos, then there exists $\delta_2>0$, such that for all $x,y\in X$ with $d(x,y)<\delta_2$,we have $|g(x)-g(y)|<\varepsilon$, for all $g\in \mathcal{G}$.
Choose $\delta=\min\{\delta_1,\delta_2\}$, such that for all $x,y\in X$ with $d(x,y)<\delta$, we have $|(f+g)(x)-(f+g)(y)|=|f(x)+g(x)-(f(y)+g(y))|=|f(x)-f(y)+g(x)-g(y)|\leq |f(x)-f(y)|+|g(x)-g(y)|<\varepsilon+\varepsilon=2\varepsilon$
Is it my work correct?
Thanks for any help.