Does the sequence still converge?

[Gargantuan]

Consider a real-valued sequence $\Big(a_i\Big)_{i=1}^{\infty}$.
Suppose that the sequence $\Big(\mid a_i \mid \Big)_{i=1}^{\infty}$ does converge
and there is a real number $K$ and a continuous function $f:[0,1]\rightarrow\mathbb{R}$ with $f(0)=0$ such that $\mid a_{n+1}-a_{n} \mid < f(\frac{1}{n})$ for all $n>K$
Does the sequence $\Big(a_i\Big)_{i=1}^{\infty}$ itself necessarily converge?

[MINGA]

It's a Cauchy sequence, so a convergent sequence.