Prove that every sequence in a discrete metric space converges and is a Cauchy sequence. This...

Prove that every sequence in a discrete metric space converges and is a Cauchy sequence.

This is all that was given to me... so I am unsure how I am supposed to prove it....

Expert Solution

Colby Messinger answered 2 years ago

prove every cauchy sequence converges

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic...

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic to a metric space (Y, dY ) of diameter at most 1.

b）Prove that every metric space is a topological space. (c) Is the converse of part (b)...

b）Prove that every metric space is a topological space. (c) Is the converse of part (b) true? That is, is every topological space a metric space? Justify your answer

Show that every cauchy sequence of reals is convergent

18.2.6. Problem. Let M be a metric space with the discrete metric. (a) Which sequences in...

18.2.6. Problem. Let M be a metric space with the discrete metric. (a) Which sequences in M are Cauchy? (b) Show that M is complete.

Prove that the discrete topology on X is the same as the metric topology induced by...

Prove that the discrete topology on X is the same as the metric topology induced by the discrete metric. Where metric topology is defined as: If (X,d) is a metric space, then consider the collection T of all open subsets of X. Then (X,T) is topological space. This topology is called the metric topology on X induced by d.

Question

Prove that every sequence in a discrete metric space converges and is a Cauchy sequence. This...

Solutions

Expert Solution

Related Solutions

prove every cauchy sequence converges

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic...

b）Prove that every metric space is a topological space. (c) Is the converse of part (b)...

Show that every cauchy sequence of reals is convergent

18.2.6. Problem. Let M be a metric space with the discrete metric. (a) Which sequences in...

Prove that the discrete topology on X is the same as the metric topology induced by...

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if...

Show that id D is dense metric space of X and if all cauchy sequences Yn...

if (a_n) is a real cauchy sequence and b is also real, prove i) (|a_n|) is...

1. Prove that any compact subset ot a metric space is closed and bounded. 2. Prove...