Is every sequence that converges to 0 also has a convergent subsequence?

[zhang128]

Homework Statement

If every subsequence of the sequence a_n converges to 0, prove that a_n itself converges to 0. Is the converse true? Prove it or give a counter example.

Homework Equations

The Attempt at a Solution

I can prove a sequence is convergent if it has a convergent subsequence. What about every sequence is convergent?

 

[VeeEight]

This is a similar reasoning to the proof that if x_n converges to x, then all subsequences that converge do so to x. You can do this by considering subsequences x_j and x_k where j is odd and k is even.

 

[rs1n]

Look at the definition of convergence. The one I learned was that a_n converges to a number L if and only if given e > 0 there exists an N such that |a_n - L| < e for all n>N. Hint: use the triangle inequality:

$$a_n - L = a_n + \underbrace{(- b_j + b_j - b_k + b_k)}_{=0} - L = (a_n - b_j) + (b_j - b_k) + (b_k - L)$$

which implies

$$|a_n-L| \le |a_n - b_j| + |b_j - b_k| + |b_k - L|$$

If you consider the b_j and b_k as subsequences, you should be able to bound the right hand side using the definition of convergence.