I don't see why this is true. For example, imagine the random variable sequence
$$(X_1 = \infty, X_1 = c, X_2 = c, \dotsc)$$
So, we have one infinity, then a whole bunch of constants. Clearly there exists another constant $k$ such that $\sum P(X_n > k)$ is finite, indeed if $k > c$, this sum equals $1 + 0 +0 + 0 + \dotsb$. So the assumption is satisfied, yet clearly supremum of this sequence is with probability $1$ equal to infinity.