Proof

Let $\sum_{k=0}^\infty a_k$ and $\sum_{k=0}^\infty b_k$ be convergent real series.
Let $c_n:=\sum_{k=0}^n a_k$ and $d_n:=\sum_{k=0}^n b_k$ be the corresponding partial sums.
The real sequences of partial sums $(c_n)_{n\in\mathbb N}$ and $(d_n)_{n\in\mathbb N}$ are convergent real sequences by hypothesis.
From the associativity and commutativity laws for adding real numbers and from the distributivity law for real numbers it follows $$\sum_{k=0}^n (a_k-b_k)=\sum_{k=0}^n (a_k+(-b_k))=\sum_{k=0}^n a_k+\sum_{k=0}^n (-b_k)=\sum_{k=0}^n a_k+(-1)\sum_{k=0}^n b_k=\sum_{k=0}^n a_k-\sum_{k=0}^n b_k=c_n-d_n.$$
By the law for the difference of convergent real sequences it follows $$\lim_{n\to\infty}(c_n-d_n)=\lim_{n\to\infty}c_n - \lim_{n\to\infty}d_n.$$
By construction, it follows $$\sum_{k=0}^\infty(a_k-b_k)=\sum_{k=0}^\infty a_k - \sum_{k=0}^\infty b_k.$$
Therefore, the real series $\sum_{k=0}^\infty (a_k-b_k)$ is a convergent real series.
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Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983