The expected number of records

Indicators. Let $I_k=1$ if the $k$-th number is a record, and $0$ otherwise. The number of records is $R=I_1+\dots+I_n$.

Probability of $I_k=1$. The $k$-th number is a record iff it is the largest among the first $k$ numbers. By symmetry, each of the first $k$ numbers is equally likely to be the largest, so
$$\Pr[I_k=1]=\frac1k.$$

Linearity.
$$\mathbb{E}[R]=\sum_{k=1}^n\frac1k=H_n.$$

Check. For $n=5$, enumerating all $120$ permutations gives mean $\frac{137}{60}=H_5$.

Remark. The indicators are even independent, so $\operatorname{Var}R=\sum\frac1k(1-\frac1k)$. Since $H_n\approx\ln n+0.577$, records are rare: among $1000$ numbers one expects about $7.5$ records. Basis of the secretary problem.

Lesson. Use symmetry to compute the probability of each indicator, then add.