Theorem: Tannery's Theorem

Let $\sequence {p_n}$ be an increasing, unbounded above sequence of natural numbers.
For every $r, n \in \N$, let $v_r$ be a mapping:
:$\map {v_r} n \in \C$
Let $\sequence {w_r}$ be a sequence of complex numbers such that, for every $r \in \N$:
:$\ds w_r = \lim_{n \mathop \to \infty} \map {v_r} n$
Let $\sequence {M_r}$ be a sequence of non-negative real numbers such that:
:$\forall n \in \N, r \le p_n: \size {\map {v_r} n} \le M_r$
and:
:$\ds \sum_{r \mathop = 0}^\infty M_r$ is convergent
Then:
:$\ds \lim_{n \mathop \to \infty} \sum_{r \mathop = 0}^{p_n} \map {v_r} n = \sum_{r \mathop = 0}^\infty w_r$

Proof:
First, observe that for all $r \in \N$:
{{begin-eqn}}
{{eqn | l = \size {w_r}
      | r = \size {\lim_{n \mathop \to \infty} \map {v_r} n}
}}
{{eqn | r = \lim_{n \mathop \to \infty} \size {\map {v_r} n}
      | c = Complex Modulus Function is Continuous
}}
{{eqn | o = \le
      | r = M_r
      | c = Lower and Upper Bounds for Sequences $(2)$
}}
{{end-eqn}}
Let $\epsilon > 0$ be arbitrary.
By Tail of Convergent Series tends to Zero, select some $q$ such that:
:$\ds \sum_{r \mathop = q}^\infty M_r < \frac \epsilon 3$
As $\sequence {p_n}$ is unbounded above, there is some $M \in \N$ such that:
:$p_M \ge q$
Since $\sequence {p_n}$ is increasing:
:$\forall n \ge M: p_n \ge p_M \ge q$
Additionally:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \paren {\sum_{r \mathop = 0}^{q - 1} \map {v_r} n - \sum_{r \mathop = 0}^{q - 1} w_r}
      | r = \lim_{n \mathop \to \infty} \sum_{r \mathop = 0}^{q - 1} \paren {\map {v_r} n - w_r}
}}
{{eqn | r = \sum_{r \mathop = 0}^{q - 1} \lim_{n \mathop \to \infty} \paren {\map {v_r} n - w_r}
      | c = Sum Rule for Complex Sequences
}}
{{eqn | r = \sum_{r \mathop = 0}^{q - 1} 0
      | c = $\ds \lim_{n \to \infty} \map {v_r} n = w_r$
}}
{{eqn | r = 0
}}
{{end-eqn}}
Therefore, we can find some $N \in \N$ such that:
:$\ds \size {\sum_{r \mathop = 0}^{q - 1} \map {v_r} N - \sum_{r \mathop = 0}^{q - 1} w_r} < \frac \epsilon 3$
Let $n \ge \max \set {M, N}$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | l = \size {\sum_{r \mathop = 0}^{p_n} \map {v_r} n - \sum_{r \mathop = 0}^\infty w_r}
      | r = \size {\sum_{r \mathop = 0}^{q - 1} \map {v_r} n - \sum_{r \mathop = 0}^{q - 1} w_r + \sum_{r \mathop = q}^{p_n} \map {v_r} n - \sum_{r \mathop = q}^\infty w_r}
      | c = $n \ge M \implies p_n \ge q$
}}
{{eqn | o = \le
      | r = \size {\sum_{r \mathop = 0}^{q - 1} \map {v_r} n - \sum_{r \mathop = 0}^{q - 1} w_r} + \size {\sum_{r \mathop = q}^{p_n} \map {v_r} n} + \size {\sum_{r \mathop = q}^\infty w_r}
      | c = General Triangle Inequality for Complex Numbers
}}
{{eqn | o = \le
      | r = \size {\sum_{r \mathop = 0}^{q - 1} \map {v_r} n - \sum_{r \mathop = 0}^{q - 1} w_r} + \sum_{r \mathop = q}^{p_n} \size {\map {v_r} n} + \sum_{r \mathop = q}^\infty \size {w_r}
      | c = Triangle Inequality for Indexed Summations and Triangle Inequality for Series
}}
{{eqn | o = \le
      | r = \size {\sum_{r \mathop = 0}^{q - 1} \map {v_r} n - \sum_{r \mathop = 0}^{q - 1} w_r} + \sum_{r \mathop = q}^\infty M_r + \sum_{r \mathop = q}^\infty M_r
      | c = $\size {\map {v_r} n} \le M_r$ and $\size {w_r} \le M_r$
}}
{{eqn | o = <
      | r = \frac \epsilon 3 + \frac \epsilon 3 + \frac \epsilon 3
}}
{{eqn | r = \epsilon
}}
{{end-eqn}}
Thus, for any $\epsilon > 0$, we have:
:$\ds \forall n \ge \max \set {M, N}: \size {\sum_{r \mathop = 0}^{p_n} \map {v_r} n - \sum_{r \mathop = 0}^\infty w_r} < \epsilon$
Thus, by definition of a limit:
:$\ds \lim_{n \to \infty} \sum_{r \mathop = 0}^{p_n} \map {v_r} n = \sum_{r \mathop = 0}^\infty w_r$
{{qed}}
{{Namedfor|Jules Tannery|cat = Tannery}}