Imma make a glossary in case someone wants to help but doesn't know the terminology.
Metric space: it's a set associated with a function d that fulfills the properties of non-negativity, commutativity and triangular inequality. The function d is called the distance.
Distance: it's the function d associated with a metric space.
Sequence: an enumerated collection of objects. Like a set, but order matters. The elements of a sequence x_n are denominated by their order x_1, x_2, x_3, and so on
Convergence: a sequence is said to be convergent when it has a limit.
Limit of a sequence: a is the limit of a sequence when for every epsilon > 0 there's a natural number n0 such that for every n > n0 you have |x_n - a| < epsilon. If a sequence doesn't have a limit, it's called a divergent sequence.
Continuity of a function in metric spaces: a function f: X |--> Y is said to be continuous when for every epsilon > 0 there's a delta > 0 such that d1(x, y) < delta implies d2(f(x), f(y)) < epsilon. In this example, x and y are any random elements of X and d1 and d2 are the distances associated with the metric spaces X and Y.
Imma make a glossary in case someone wants to help but doesn't know the terminology.
Metric space: it's a set associated with a function d that fulfills the properties of non-negativity, commutativity and triangular inequality. The function d is called the distance.
Distance: it's the function d associated with a metric space.
Sequence: an enumerated collection of objects. Like a set, but order matters. The elements of a sequence x_n are denominated by their order x_1, x_2, x_3, and so on
Convergence: a sequence is said to be convergent when it has a limit.
Limit of a sequence: a is the limit of a sequence when for every epsilon > 0 there's a natural number n0 such that for every n > n0 you have |x_n - a| < epsilon. If a sequence doesn't have a limit, it's called a divergent sequence.
Continuity of a function in metric spaces: a function f: X |--> Y is said to be continuous when for every epsilon > 0 there's a delta > 0 such that d1(x, y) < delta implies d2(f(x), f(y)) < epsilon. In this example, x and y are any random elements of X and d1 and d2 are the distances associated with the metric spaces X and Y.