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- Prove the
*Triangle inequality*:

If*a*,*b*are real numbers then |*a*+*b*| |*a*| + |*b*|.

When is this inequality strict ?

Why is it called the Triangle inequality ?

Use induction to prove that if*a*_{1},*a*_{2}, ...,*a*_{n}are any real numbers then |*a*_{i}| |*a*_{i}|. - Prove the
*Squeeze rule*:

If*a*_{n}*x*_{n}*b*_{n}for all*n*and the sequences (*a*_{n}) and (*b*_{n}) converge to the same limit, then (*x*_{n}) also converges to this limit. - Give examples or prove the non-existence of sequences which are:

(a) convergent but not monotonic,

(b) bounded but not convergent,

(c) convergent but not bounded,

(d) monotonic but not bounded,

(e) divergent to + but not monotonic,

(f) monotonic and bounded but not convergent,

(g) unbounded but not monotonic,

(h) positive and convergent, but not monotonic. - The sequence ( (2
*n*+ 1)/(*n*+ 1) ) converges to 2. How big must we choose*N*to be so that the terms of the sequence are within of this limit if = 0.1, 0.01, 0.0001 ?Do the same thing for the sequence ( (2

*n*^{2}+ 1)/(*n*^{2}+1) ). - Give examples or prove the non-existence of sequences such that:

(a) (|*a*_{n}|) converges but (*a*_{n}) does not converge,

(b) (*a*_{n}) converges but (|*a*_{n}|) does not converge,

(c) (*a*_{n}) and (*b*_{n}) do not converge, but (*a*_{n}+*b*_{n}) converges,

(d) (*a*_{n}) and (*a*_{n}+*b*_{n}) converge but (*b*_{n}) does not converge. - Let
*a*be a real number in (0, 1). Let*a*_{n}be the real number obtained by deleting the first*n*digits of the decimal expansion of*a*.

For example, if*a*= 0.1234567... then*a*_{1}= 0.234567..,*a*_{2}= 0.34567..., etc.

For which*a*is the sequence (*a*_{n}) convergent and what is its limit ?

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