Showing convergent sequence is Cauchy

[patter2809]

hi, sorry if this is the wrong section in which to post this!

Q. A sequence (a_n)_n∈ ℕ is said to be Cauchy if for every ε > 0, there exists an N₀ ∈ ℕ such that for all n, m > N, |a_n-a_m| < ε.

Show that every convergent sequence is Cauchy.

A so far. Let ε be such that |a_N0 - L| < ε which means that we certainly have |a_n - L| < ε, and |a_m - L| < ε, since n, m > N₀.

|a_n - a_m| = |a_n - L + L - a_m​|
≤ |a_n - L| + |a_m - L| < ε + ε = 2ε

Not sure how to proceed from here.

 

[pka]

Q. A sequence (a_n)_n∈ ℕ is said to be Cauchy if for every ε > 0, there exists an N₀ ∈ ℕ such that for all n, m > N, |a_n-a_m| < ε.
Show that every convergent sequence is Cauchy.

A so far. Let ε be such that |a_N0 - L| < ε which means that we certainly have |a_n - L| < ε, and |a_m - L| < ε, since n, m > N₀.
|a_n - a_m| = |a_n - L + L - a_m​|
≤ |a_n - L| + |a_m - L| < ε + ε = 2ε

All of that is correct. But had you used \(\displaystyle \dfrac{\epsilon }{2}\) in the first place you would end with \(\displaystyle |a_n-a_m|<\epsilon\)

 

[patter2809]

ah, had a feeling i was being stupid. thank you very much!