Defining a sequence with contstraints

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Good morning,

I have difficulties to solve the following problem. I need to find a decreasing sequence, say $(c_t)_t$ (the starting point of the sequence is not important), such that:

1) $\lim_{t}c_t=0$;

2) $c_t - c_{t+1}\geq \frac{1}{log(t+1)t}$ for all $t$.

Does such a sequence exists? If so, is it possible to provide an example?

Thanks in advance,

Umberto