# Sequence notation

We often want to prove that there exists a sequence of values , all in , satisfying some property. Formally, we would say "there exists and values such that ".

This takes a lot of writing, and also requires us to introduce the variable (which often just adds complexity). So, we will abbreviate this to "there exists such that ".

We will also abbreviate sums and products of all the values: denotes the sum of the and 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.