# Different proofs of $\lim_{x\to \infty}\left(1+ \frac{1}{n}\right)^n =e$ [duplicate]

I recently was teaching my friend about the number $e$. I introduced him the number by using the compound interest thing . Then I wrote down the general result -$$\lim_{x\to \infty}\left(1+ \frac{1}{n}\right)^n =e$$ The he told me that yes it works for $n=10,100,200,1000$. Beyond that his computer couldn't check . So he asked me for a formal mathematical proof of it . I thought of one but then that proof had natural logarithms - meaning they involved the number $e$ .I want to know the different ways through which this results can be proved , but without any use of $e$.

## marked as duplicate by Guy Fsone, mechanodroid, user99914, Claude Leibovici limits StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Nov 11 '17 at 6:06

• How can you prove the result "without any use of $e$" if the answer is itself $e$? – 6005 Dec 17 '13 at 9:13
• How do you define $e$? Afaik, it's normally defined as the limit you want to prove. – Daniel R Dec 17 '13 at 9:13
• Does he have a VIC 20? Google can calculate the value (approximately of course) for $n=100000000$ (and much higher values, I'm sure). – David Mitra Dec 17 '13 at 9:17