How can I plot out the mean?

I'm trying to using simulations, draw 1 to 1000 samples from a population with a mean of 50 and a standard deviation of 10. Calculate the mean of each sample, and make a plot that shows how that mean changes as you increase your sample size. Based on that plot, is the mean a biased or an unbiased estimate of the population mean?

Below is what I have done:

sd_uncorrected<-function(x){ return(sqrt(sum((x-mean(x))^2)/length(x)))
}

population <- rnorm(n = 1000, mean = 50, sd = 10) 
population_mean <- mean(population) 
population_std <- sd_uncorrected(population)

paste('population mean=',population_mean)
paste('population std = ', population_std)

sample_size <- 1000 # how many elements we want to sample
sample_n <- sample(population, size = sample_size, replace = FALSE) 
sample_n

mean(sample_n)
sd_uncorrected(sample_n)

n_experiments <- 1000 # we will sample 1000 times
sample_size <- 10 # how many elements we want to sample?
sample_means <- c()

library(ggplot2)
sample_means_df <- data.frame(means=sample_means)
ggplot(sample_means_df, aes(x=means)) + geom_histogram() +
geom_vline(xintercept = population_mean, color='red') + # population mean
geom_vline(xintercept = mean(sample_means_df$means), color='black')

I'm getting the follow error message and I don't know what I need to do. Can someone please help me?

Error in FUN(X[[i]], ...) : object 'means' not found
In addition: Warning message:
In mean.default(sample_means_df$means) :
argument is not numeric or logical: returning NA