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Given ${\{a_n}\}$ and ${\{b_n}\}$ such that $0\le{a_n}\le{b_n}$:

if $\sum_{n=0}^{+\infty}b_n$ converges than $\sum_{n=0}^{+\infty}a_n$ does,

if $\sum_{n=0}^{+\infty}a_n$ diverges positively than $\sum_{n=0}^{+\infty}b_n$ does too.

But I don't get why, given $r_n=\sum_{k=n+1}^{+\infty}a_k$ and $R_n=\sum_{k=n+1}^{+\infty}b_k$ it is (eventually) $r_n\le R_n$

Can you help me with a demonstration and possibly a visual representation? Thanks.

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Summing a finite number of inequalities $a_k\le b_k$: $$\sum_{k=n+1}^m a_k\le\sum_{k=n+1}^m b_k.$$ And now, take $\lim_{m\to\infty}$

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