If an equality is true for any finite N then we can take the limit

Question

Suppose we have the following equality.

$\mu (F_{N})= \sum_{i=0}^{N}G_{n}$ for any finite $N$.

Is it true that we can then the limit if this relation then? i.e

lim $\mu (F_{N})=\sum_{i=0}^{\infty}G_{i}$

Answer 1

Suppose you have an equality $a_n = b_n$ that holds for all $n\in\mathbb N$. If (without restriction of generality) there exists $a = \lim_{n\to\infty} a_n$ then we can simply see that

$$ a = \lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n, $$ because every $a_n$ is equal to $b_n$.

Therefore we know that, if we have an equality that depends on a variable, we can simply apply the limit to both sides and still have an equality.

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Concerning your problem in particular: the series $\sum_{i=0}^\infty G_i$ is called convergent if the sequence of partial sums converges: $\sum_{i=0}^\infty G_i := \lim_{N\to\infty}\sum_{i=0}^N G_i$.