Prove that $\displaystyle \sum_{n=1}^\infty\!a_n $ converges if and only if $\displaystyle \sum_{n=N}^\infty\!a_n $ converges for any $\displaystyle N \geq 1 $

My approach..

Assume that the latter series converges then this inequality holds for some real $\displaystyle M $

$\displaystyle \left| \sum_{n=1}^\infty\!a_n - \sum_{n=N}^\infty\!a_n \right| < M $ for any $\displaystyle N \geq 1 $

And thus does also the first series converge. They only differ by a constant depending on $\displaystyle N $.

Now assume that $\displaystyle \sum_{n=N}^\infty\!a_n $ diverges to $\displaystyle \pm\infty$ for some $\displaystyle N > 1 $ and that $\displaystyle \sum_{n=1}^\infty\!a_n $ converges.

$\displaystyle \sum_{n=1}^\infty\!a_n = \sum_{n=1}^{N-1}a_n + \sum_{n=N}^\infty\!a_n$

This inequality holds for some $\displaystyle M $

$\displaystyle \left| \sum_{n=1}^\infty\!a_n - \sum_{n=1}^{N-1}\!a_n \right| < M $ for any $\displaystyle N \geq 1 $

But this inequality doesn't hold for any $\displaystyle M $

$\displaystyle \left| \sum_{n=N}^\infty\!a_n \right| < M $ for any $\displaystyle N \geq 1 $

Contradiction!

My assumtition must be wrong.

Either does both diverge to $\displaystyle \pm\infty $ or converge.

Is there something wrong? Something missing to complete the proof?