Question

Show that a sequence {sn} coverages to a limit L if and only if the sequence {sn-L} coverages to zero.

Answer 1

Explanation:

To prove this we can use the definition of a sequence converging to its limit, in terms of epsilon:

The sequence
`\{ S_n\}` converges to
`L`

if and only if

for every
`\epsilon >0` there exists
`n_0\in \mathbb{N}` such that

`n>n_0 \implies |S_n-L|<\epsilon`

if and only if

for every
`\epsilon >0` there exists
`n_0\in \mathbb{N}` such that
`n>n_0 \implies |(S_n-L) - 0|<\epsilon`

if and only if

the sequence
`\{S_n-L\}` converges to 0.