If the difference between two consecutive elements tends to 0, then the sequence converges? Mistake in proof

I tried proving that if $a_{n+1}$ - $a_n$ $\rightarrow$ $0$, then $a_n$ converges.
Now I searched for this example here and saw many counterexamples, yet I don't understand where my proof went wrong.

Proof:

Proof by induction that $a_n$ is Cauchy.
base:
for $k=1$ we know that $|a_{n+k} - a_n|< \epsilon$ for every $\epsilon$ > $0$.

Step:
assume this condition is true for $k$ and prove for $k+1$, then:
$$|a_{n+k+1}-a_n| = |a_{n+k+1}-a_k+a_k+a_n| \leq |a_{n+k+1}-a_k| + |a_k-a_n| \leq 2\epsilon$$ So by this point can't I say than $a_n$ converges from Cauchy?

The only thing I could think about was that maybe the induction doesn't cover the Cauchy condition that says this needs to be $\forall m,n \geq N$ but I never encountered this so I can't really trust my reasoning.

Solution 1:

Your base condition is not a correct statement of what it means for $\ a_{n+1}-a_n\ $ to converge to zero. A correct statement is:

For any $\ \epsilon>0\ $ there is a positive integer $\ N\ $ such that $\ \big|\,a_{n+1}-a_n\,\big|<\epsilon\ $ for all $\ n\ge N\ $.

With this as the induction base, the statement you 'd be trying to prove (and can, in fact, prove) by induction for all positive integers $ \ k\ $ is

For any $\ \epsilon>0\ $ there is a positive integer $\ N\ $ such that $\ \big|\,a_{n+k}-a_n\,\big|<\epsilon\ $ for all $\ n\ge N\ $.

But this being true for all $\ k>0\ $ is not the Cauchy criterion for convergence; it is merely a statement that $\ \,a_{n+k}-a_n\rightarrow0\ $ for all positive integers $\ k\ $.

I suspect you're confusing the above-quoted statement with the following version of Cauchy's criterion

For any $\ \epsilon>0\ $ there is a positive integer $\ N\ $ such that $\ \big|\,a_{n+k}-a_n\,\big|<\epsilon\ $ for all $\ n\ge N\ $ and all positive integers $\ k\ $.

It might help you see the difference between these statements by expressing them with symbolic quantifiers. The first is $$ \forall k\in\mathbb{N}\,\forall\epsilon\in\mathbb{R}_{>0}\,\exists N\in\mathbb{N}\,\forall n\in\mathbb{N}, n\ge N\,\big|\,a_{n+k}-a_n\,\big|<\epsilon\ , $$ whereas the version of Cauchy's criterion given above is $$ \forall\epsilon\in\mathbb{R}_{>0}\,\exists N\in\mathbb{N}\,\forall n\in\mathbb{N}, n\ge N\,\forall k\in\mathbb{N}\ \big|\,a_{n+k}-a_n\,\big|<\epsilon\ . $$ In the first of these statements the existential quantifier over $\ N\ $ appears after the universal quantifier over $\ k\ $ which means that $\ N\ $ can, and generally will, depend on the value of $\ k\ $. In the Cauchy criterion, however, the existential quantifier over $\ N\ $ appears before the universal quantifier over $\ k\ $, which makes it impossible for $\ N\ $ to depend on $\ k\ $.