Every definition of consistency I see mentions something convergence in probability-like in its explanation.
From Wikipedia's definition of consistent estimators:
having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to $θ_0$.
That sounds like consistency implies convergence in probability. Though, I never see it said so plainly.
Likewise, from this discussion of unbiasedness and consistency:
Roughly speaking, consistency means that for large values of 𝑛 we are going to be close to the true value of the parameter with a high probability
Is it the case that estimator consistency $\iff$ convergence in probability? Consistency seems somewhat informally defined, and I'm not seeing it stated mathematically.