Prove that the first few terms of a sequence do not affect convergence. Formally, show that if there exists an N such that an = bn for n > N then, an —»• a as n —»• oo implies bn —> a as n —> oo.

I shall give the proofs in detail but the reader is warned that similar proofs will be left to her in the remainder of the book.
(i) By definition:-

We work in an ordered field F. We say that a sequence a\, a,2, • • • tends to a limit a as n tends to infinity, or more briefly
an —»• a as n —»• oo
if, given any e > 0, we can find an integer no (б) [read 'щ depending on e'] such that
\an — a\ < e for all n > no (б).
The following properties of the limit are probably familiar to the reader.

Many ways have been tried to make calculus rigorous and several have been successful. We choose the first and most widely used path via the notion of a limit. In theory, my account of this notion is complete in itself. However, my treatment is unsuitable for beginners and I expect my readers to have substantial experience with the use and manipulation of limits.|