The random variable $T_i$, the “Hitting Time of $i$” is defined to be the first $n$ such that $X_n=i$ given that $X_0=i$.

By the mean recurrence time of $T_i$, I mean the expected value of this random variable.

I wish to show that if $i$ is transient, then the expectation does not converge to any finite real number. While this, intuitively makes sense, I do not know how to formally prove this and any help is appreciated.