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**charikaar** Suppose that $\displaystyle (x_k)_{k=1}^{\infty}$ and $\displaystyle (y_k)_{k=1}^{\infty}$ are sequences of positive real numbers, and that $\displaystyle \sum_{k=1}^{\infty}x_{k}=S$ and $\displaystyle \sum_{k=1}^{\infty}y_{k}=T$. Let $\displaystyle z_k=\max(x_k,y_k)$. Prove that the sum $\displaystyle \sum_{k=1}^{\infty}z_k$ exists.

Let $\displaystyle S_n=\sum_{k=1}^{\infty}x_{k}$ and let

Let $\displaystyle T_n=\sum_{k=1}^{\infty}y_{k}$

We know $\displaystyle S_n$ coverges to S and $\displaystyle T_n$ converges to T. So $\displaystyle S_n+T_n$ converges to S+T. Am i on the right track? I can't go any further from here so any help would be greatly appreciated.

Thanks