So, I proved that $\big((1+\frac{x}{n})^n\big)_{n\in N}$ converges for $x\in$ $[0,1]$ (call it fact (1)) and that $\lim_{n\to \infty} |(1+\frac{a}{n})^n - (1+\frac{a}{n} + \frac{b}{n^2})^n| = 0$ for any $a, b\in$ R (call it fact (2)).
Now, the question I'm solving requires me to prove the convergence of $\big((1+\frac{x}{n})^n\big)_{n\in N}$ for all real numbers using specifically these two facts. Both of them.
Some ideas and tools I considered using, but couldn't make it work were the Binomial Expansion and using convergence tests as the comparison test. More specifically, I considered comparing the above sequence with $\big((1+\frac{1}{n})^n\big)_{n\in N}$ .
Any help appreciated.