Let series $\displaystyle \sum _{n=n_0}^{+\infty}(-1)^na_n$ satisfy Leibniz convergence criterion. Let $\displaystyle r_n=\sum_{k=n+1}^{+\infty}(-1)^ka_k$. Prove that $r_n$ has the same sign as its first term and |r_n|
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Well, so far I used this fact several times but never consider proving it. In my opinion proofs of simple facts are not so obvious. I don't know how to precise show it.