The sample mean is unbiased

By definition, $\bar X = \frac{1}{n}\sum_{i=1}^{n} X_i$. Expectation is a linear operator, so it distributes over the sum and the constant factor $\frac{1}{n}$: $E[\bar X] = \frac{1}{n}\sum_{i=1}^{n} E[X_i]$.

Since every $X_i$ comes from the same population, $E[X_i] = \mu$ for each of the $n$ terms, so the sum collapses to $\frac{1}{n}\sum_{i=1}^{n} \mu = \frac{1}{n}(n\mu) = \mu$.

Therefore $E[\bar X] = \mu$ for every sample size $n$: the sample mean is unbiased no matter how small or large the sample is. This is a statement about the center of its sampling distribution only -- the spread of that distribution, $\operatorname{Var}(\bar X) = \sigma^2/n$, still shrinks as $n$ grows, which is what the confidence interval below uses.