Let $\displaystyle \sum_{n=0}^{\infty} a_{n} $ be a convergent series of non-negative real numbers, and let $\displaystyle f: \bold{N} \to \bold{N} $ be a bijection. Prove that $\displaystyle \sum_{m=0}^{\infty} a_{f(m)} $ is also convergent, and has the same sum: $\displaystyle \sum_{n=0}^{\infty} a_{n} = \sum_{m=0}^{\infty} a_{f(m)} $.

Is this the general idea of the proof: Let the partial sums be $\displaystyle S_{N} := \sum_{n=0}^{N} a_{n} $ and $\displaystyle T_{M} := \sum_{m=0}^{M} a_{f(m)} $. Then write $\displaystyle L := \sup(S_{N})_{n=0}^{\infty} $ and $\displaystyle L' = \sup(T_{M})_{m=0}^{\infty} $. Show that $\displaystyle L = L' $ or in other words, show that the sums are equal? And to show equality, we show the following two inequalities to hold simultaneously: $\displaystyle L \leq L' $ and $\displaystyle L' \leq L $?