# Every closed ball in X is complete, then X is complete

Homework Statement .
Prove that if every closed ball in a metric space X is a complete subspace, then X is complete.

The attempt at a solution.

Let ##\{x_n\}## be a cauchy sequence in X. Then, for ##ε=1##, ##\exists## ##n_0 \in \mathbb N##: ##\forall## ##m≥n_0##,##n≥n_0## ##d(x_m,x_n)<1##. Consider the closed ball centered at ##x_{n_0}## of radius 1. Then, the subsequence ##\{x'_n\}## such that ##x'_n=x_{n_0 + n-1}## is a subsequence of ##\{x_n\}## included in ##K(x_{n_0},1)## . By hypothesis, the ball is a complete subspace and the subsequence ##\{x'_n\}## is cauchy. But then ##\{x'_n\}## converges to some point ##x \in K(x_{n_0},1)## This implies ##\{x'_n\}## converges to the same point x in the metric space X. So the original sequence ##\{x_n\}## has a convergent subsequence that converges to the point x, this means that ##\{x_n\}## converges to x. I've proved that an arbitrary cauchy sequence in X is convergent, this implies every cauchy sequence in X is convergent, then X is a complete metric space.

Is my proof correct?

Dick
Homework Helper
Homework Statement .
Prove that if every closed ball in a metric space X is a complete subspace, then X is complete.

The attempt at a solution.

Let ##\{x_n\}## be a cauchy sequence in X. Then, for ##ε=1##, ##\exists## ##n_0 \in \mathbb N##: ##\forall## ##m≥n_0##,##n≥n_0## ##d(x_m,x_n)<1##. Consider the closed ball centered at ##x_{n_0}## of radius 1. Then, the subsequence ##\{x'_n\}## such that ##x'_n=x_{n_0 + n-1}## is a subsequence of ##\{x_n\}## included in ##K(x_{n_0},1)## . By hypothesis, the ball is a complete subspace and the subsequence ##\{x'_n\}## is cauchy. But then ##\{x'_n\}## converges to some point ##x \in K(x_{n_0},1)## This implies ##\{x'_n\}## converges to the same point x in the metric space X. So the original sequence ##\{x_n\}## has a convergent subsequence that converges to the point x, this means that ##\{x_n\}## converges to x. I've proved that an arbitrary cauchy sequence in X is convergent, this implies every cauchy sequence in X is convergent, then X is a complete metric space.

Is my proof correct?

Why are you having doubts? Eventually the tail of every cauchy sequence is contained in a closed ball. The closed ball is complete. So the sequence is convergent.

• 1 person
Why are you having doubts? Eventually the tail of every cauchy sequence is contained in a closed ball. The closed ball is complete. So the sequence is convergent.

Thank you, I'm new to proofs and mathematics in general so I usually have strong doubts on whether I am doing the exercises right or not. I suppose it's a newbie's insecurity.

Dick
Homework Helper
Thank you, I'm new to proofs and mathematics in general so I usually have strong doubts on whether I am doing the exercises right or not. I suppose it's a newbie's insecurity.

Well, I thought that was a pretty fine and clear proof. I think you should start feeling less insecure.

The only thing is that you don't want to say ε=1.

Because your metric space may be "smaller" than the unit closed ball.

You could just say for some ε. But the proof seems fine

Dick