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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!

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

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