# Generalizing a concept about diverging sequences

I just started learning Analysis and I'm trying to generalize a concept that I have seen but never proved.

If $$\alpha_n$$ is some sequence in $$\mathbb{R}$$, that can be written as the sum $$\alpha_n=\beta_n+\gamma_n$$, then we say that $$\alpha_n$$ diverges if $$\beta_n$$ converges and $$\gamma_n$$ diverges.

First of all, is this property even true?

If it is true, how would I prove it rigorously? I can't see how I would apply the definition of the convergence of a sequence here.

All help is appreciated!

• Hint: if $\alpha_n$ converged, so would $\alpha_n-\beta_n$. – Wojowu Feb 13 at 20:22