Sequence notation

From CS2800 wiki

We often want to prove that there exists a sequence of values [math]a_0, a_1, \dots, a_k [/math], all in [math]X [/math], satisfying some property. Formally, we would say "there exists [math]k \href{/cs2800/wiki/index.php/%E2%88%88}{∈} \href{/cs2800/wiki/index.php/%E2%84%95}{ℕ} [/math] and values [math]a_0, a_1, \dots, a_k \href{/cs2800/wiki/index.php/%E2%88%88}{∈} X [/math] such that [math]\dots [/math]".

This takes a lot of writing, and also requires us to introduce the variable [math]k [/math] (which often just adds complexity). So, we will abbreviate this to "there exists [math](a_i) \href{/cs2800/wiki/index.php/%E2%88%88}{∈} X [/math] such that [math]\dots [/math]".

We will also abbreviate sums and products of all the values: [math]\sum_i a_i [/math] denotes the sum of the [math]a_i [/math] and [math]\prod_i a_i [/math] denotes their product. We won't worry too much about the indices; unless otherwise specified, we just mean add (or multiply) all of them.

Note: there is ambiguity in this notation about whether we allow finite or infinite sequences. I will only use this notation for finite sequences.